math 301, 1.1 ancient mathematics name: reading...

Math 301, 1.1 Ancient Mathematics

Name:

Reading Questions

1. Explain the geometric proof of the quadratic formula.

2. The door problem was created to teach students how to use what two facts/theorems?

3. Pick your two favorite odd numbers and use them to generate a Pythagorean triple using the

Babylonian method.

4. What struck you in reading this section? Any points of confusion?

1

Math 301, 1.2 Diophantus-I

Name:

Reading Questions

1. Using the parametrization in Prop. 1.2, what is the parameter t corresponding to the point

P = (

p32 , 1

2 ) on the unit circle? What point on the unit circle corresponds to a t-value of 1?

2. Give the definition of Pythagorean point, and draw a picture to illustrate. Come up with three

examples, using Plimpton 322 (Figure 1.5 in Section 1.1) and Prop. 1.3.

3. Draw a picture of the unit circle and the two lines corresponding to t =p2� 1 and t = 1 in Thm. 1.4.

1

Math 301, 1.2 Diophantus-I

4. Do exercise 1.19(i). (Warning: the answer given in the text is incorrect.)

5. What is a primitive Pythagorean triple?

6. What struck you about this section? What was confusing?

2

Math 301, 1.2 Diophantus-II

Name:

Reading Questions

1. Give the statement of Fermat’s Last Theorem (FLT). Why is it named after Fermat? Did he prove it?

2. What is the “method of infinite descent”?

3. Explain, in your own words, why FLT would be true if every odd prime were “good.”

4. What is the definition of a congruent number?

1

Math 301, 1.2 Diophantus-II

5. Pick a Pythagorean triple from Plimpton 322 and use it to find a congruent number, using Prop. 1.10.

6. What struck you in this section? What part was the most unclear or confusing?

2

Math 301, 1.3 Euclid-I

Name:

Reading Questions

1. State the Least Integer Axiom. If the words “natural numbers” were replaced with “integers,” would

the statement still be true? Explain.

2. Do 1.3.37.

3. What is the restriction on the remainder in the Division Algorithm? Why is it important?

4. State Theorem 1.19.

1

Math 301, 1.3 Euclid-I

5. State the definition of gcd, and give an equivalent condition (Corollary 1.20).

6. State the definition of prime, and give an equivalent condition (Theorem 1.21, Euclid’s Lemma).


2

Math 301, 1.3 Euclid-II

Name:

Reading Questions

1. Find gcd(480, 126) using antanairesis as in the first paragraph on page 31.

2. Use the Division Algorithm and Lemma 1.27(i) to describe your work above more e�ciently, as in the

second paragraph on page 31.

3. Rewrite your work above in the notation of the proof of Theorem 1.29 as in the paragraph after the

proof.

1

Math 301, 1.3 Euclid-II

4. Find coe�cients s and t expressing gcd(480, 126) as a linear combination of 480 and 126, as in the

paragraph after the proof of Theorem 1.30.


2

Math 301, 1.4 Nine Fundamental Properties

Name:

Reading Questions

1. List the nine fundamental properties (with names, applicable.)

2. Does Q satisfy all nine properties? Does Z? Does N? If not, which properties are not sastisfied?


Math 301, 2.1-I Induction

Name:

Reading Questions

1. Give an example of a failure of inductive reasoning.

2. Why is inductive reasoning valuable in mathematics?

3. How is mathematical induction di↵erent from inductive reasoning?

4. Give an example of an implication that is true but whose conclusion is false. (See “How To ThinkAbout It” box, p 49.)

5. What struck you? What is unclear to you in this reading? What questions do you have?

Math 301, 2.1-II Fundamental Theorem of Arithmetic

Name:

Reading Questions

1. State the Fundamental Theorem of Arithmetic and its corollary.

2. Find the prime factorizations of the numbers 480 and 126. Write the prime factorizations in the formgiven in Theorem 2.10 and in the form given in Corollary 2.11.

3. Find the lcm and gcd of 480 and 126 using prime factorization, as described in Proposition 2.13. Howdoes the form of the prime factorization in this proposition di↵er from that of Corollary 211?

4. Verify Corollary 2.14 in the case of a = 480 and b = 126.

Math 301, 2.1-II Fundamental Theorem of Arithmetic

5. How does strong induction di↵er from induction? When is it advantageous?


Math 301, 2.2 Binomial Theorem

Name:

Reading Questions

1. Verify that the sixth row of Pascal’s triangle can be obtained from the fifth row, as described on the

top of p 64.

2. State Pascal’s Theorem (Proposition 2.24).

3. Verify that the numbers in the sixth row of Pascal’s triangle satisfy the formula given in Pascal’s

Theorem.


Math 301, 3.1 Classical Formulas

Name:

Reading Questions

1. Verify that the three cube roots of unity (1, !, and !̄) are solutions of x

3 � 1 = 0.

2. What are the three cube roots of 8?

3. When we are interested in solutions to the general cubic equation, aX

3+ bX

2+ cX + d = 0, it

su�ces to consider the case a = 1, b = 0. Why is this su�cient?

4. In what sense is Example 3.4 “good” and Example 3.5 “bad”?

Math 301, 3.1 Classical Formulas

5. True or false: the procedures for finding roots of the general cubic and the general quartic can be

extended to find roots of the general quintic.


Math 301, 3.2-I Complex Numbers

Name:

Reading Questions

1. Sum up Proposition 3.8 in one sentence. (See the text immediately preceding the proposition.)

2. Find the reciprocal of the complex number 3 + 4i using the formula on the top of page 96, and check

your answer by multiplying by 3 + 4i to get 1.

3. Find the reciprocal of 3 + 4i using the complex conjugate, as in Proposition 3.11.

4. Cite the relevant parts of propositions in this section that correspond to the nine fundamental

properties of the real numbers described in Section 1.4. (Include the names of the properties.)


Math 301, 3.2-II Complex Numbers

Name:

Reading Questions

1. A vector has magnitude and direction. What are the corresponding geometric properties of a complex

number? (List multiple names for the same property, if multiple names are given.)

Given a complex number z, how does one find these two properties?

2. Describe the geometric interpretation of multiplication of complex numbers in one sentence.

3. Let z =

p3 + i and w = � 3

2 +

3p3

2 i. Find zw in two ways: (a) using the definition of multiplication

on page 94 and (b) converting z and w to polar form and using Theorem 3.18.

Math 301, 3.2-II Complex Numbers

Recall: z =

p3 + i and w = � 3

2 +

3p3

2 i. Above you found zw in two ways. Verify that the two are

the same by converting your result from (a) to polar form. Illustrate with a picture of z, w and zw in

the complex plane.

Consider an arbitrary nonzero complex number w0, and let z be as above. Where is zw0

located in

the complex plane, relative to w0? Illustrate with a picture.


Math 301, 3.3 and 3.4 Roots and Powers, Norms, Gaussian and Eisenstein Integers

Name:

Note Read all of Section 3.3, and read the following three parts of Section 3.4: Norms (p 116-117),

Gaussian Integers: Pythagorean Triples Revisited (p 119-120), and Eisenstein Integers (p 120-121 up to

but not including the definition of an Eisenstein triple).

Reading Questions

1. State Euler’s theorem (Theorem 3.25).

2. State De Moivre’s Theorem, in both forms (Theorem 3.20 and Corollary 3.27).

3. Write the complex roots of x

12 � 1 in exponential polar form, and draw them in the complex plane.

Which of these are primitive?

What is �(12)?

If ⇣ denotes one of the complex roots of x

12 � 1, what is ⇣

49?

Math 301, 3.3 and 3.4 Roots and Powers, Norms, Gaussian and Eisenstein Integers

4. State the definition of the norm of a complex number.

5. State the definitions of the Gaussian integers and the Eisenstein integers.


Math 301, 4.1 Congruence

Name:

Read the introduction to Chapter 4 (the first few paragraphs on page 131) and Section 4.1, up to but notincluding Proposition 4.11 on page 137.

Reading Questions

1. The introduction to Chapter 4 provides the context and motivation for what we will study inChapter 4. Describe this “big picture” in your own words. How does Chapter 4 fit in with what wehave studied and what we will study?

2. The first few paragraphs of Section 4.1 provide the motivation for studying congruence. Describe themotivation in your own words.

3. The next several pages give the definition of congruence modulo m and state several results necessaryfor understanding arithmetic modulo m (Proposition 4.2, Proposition 4.3, Corollary 4.4, andProposition 4.5.) Write the definition of congruence modulo m in your notebook (word for word),and take notes on the propositions. Which of these are confusing to you?

4. Example 4.1 illustrates the idea of congruence modulo m in daily life. Use this example or anexample of your own to understand the statements of the propositions and corollary. Take some notes(in your notebook) on the examples that help you to understand the points that were confusing toyou originally. Give an example or two of your own below.

Math 301, 4.1 Congruence

5. Next there are several examples illustrating the usefulness of congruence for solving a variety ofproblems. Reread Example 4.6 carefully, and try the example proposed by the author at the end ofthe example.

6. The next three results (Proposition 4.8, Theorem 4.9, and Corollary 4.10) are all related and will beuseful in an application (what is the application?) Which of these are unclear to you? Try verifyingthem with a few examples.

7. What struck you in this reading? What is still unclear? What remaining questions do you have?

Math 301, 4.1-II and 4.2, Linear Congruences and Public Key Codes

Name:

Read pages 141-143 of Section 4.1 (Linear Congruence and the Chinese Remainder Theorem) and read all

of Section 4.2 (Public Key Codes). There are a lot of details to keep track of - especially in section 4.2 - so

read carefully with paper and pencil.

Reading Questions

1. Under what conditions is the linear congruence ax ⌘ b mod m solvable?

2. Example 4.19 illustrates the method described in the proof of Theorem 4.17 for solving linear

congruences. Reread Example 4.19 carefully to make sure you understand all the steps. Now create

your own example of a linear congruence, and solve it. (Make sure that your a, b, and m satisfy the

hypotheses in the theorem so that your linear congruence is solvable.)

3. The Chinese Remainder Theorem deals with solutions of systems of linear congruences. Copy that

theorem in your notebook.

4. Create your own example of a system of linear congruences that the Chinese Remainder Theorem

would apply to. (Choose m, m

0, b, b

0satisfying the hypotheses of the theorem, and write out the

system of linear congruences.) What does the theorem allow you to conclude? (State the conclusion

of the theorem in terms of your particular choices of m and m

0.) You do not need to solve the

system.

Math 301, 4.1-II and 4.2, Linear Congruences and Public Key Codes

5. Section 4.2 describes the ingredients of RSA codes. Record the description of each component,

explain how they are related, and explain how they are used.

6. Reread Example 4.28, and outline the main steps used to encode and decode the given word.


Math 301, 4.3 Commutative Rings-I: Zm and Abstract Commutative Rings

Name:

Read Section 4.3, up to but not including Units and Fields, page 160.

Reading Questions

1. The section begins by introducing the integers mod m, denoted Zm. (Write this definition, word for

word, in your notebook.) The text illustrates the main points using Z2 as an example. Pick your

favorite integer (between 3 and 7), and reread this part of the section with your example in mind.

Make addition and multiplication tables for your example (like the second pair of tables near the top

of page 155.)

2. Next is the definition of a commutative ring. (Write this definition, word for word, in your notebook.

Make sure not to leave out important phrases like, “for all” and “there exists.”)

3. Following the definition are several examples, several of which we have already studied. Pay

particular attention to the examples in Example 4.30 and Theorem 4.32.

4. Reread the first paragraph under Properties of Commutative Rings (page 159), and scan the

statements and proofs of the results (namely the propositions, corollaries, and theorem) on the next

two pages. Summarize the paragraph in your own words.

Math 301, 4.3 Commutative Rings-I: Zm and Abstract Commutative Rings

5. Carefully reread the two definitions on the top of page 160. In what sense are these notations

“hybrid” notations?


Math 301, 4.3 Commutative Rings-II: Units and Fields

Name:

Read Section 4.3, pages 160-164, Units and Fields. This is a short reading assignment, so take the time to

read it carefully.

Reading Questions

1. There are just two definitions in this reading assignment. What are they? Find them and write them

in your notebook, word for word.

2. Reread Example 4.40. Are any of these unclear? If so, take the time to verify the examples for

yourself in your notebook.

Pick your favorite prime p, and list the units in Zp.

Pick your favorite composite (i.e. not prime) integer m between 10 and 20, and list the units in Zm.

3. Study the proofs of Proposition 4.39 and Theorem 4.43, making sure to look up the references to

previously proven results. Are there any steps that are unclear?

Math 301, 4.3 Commutative Rings-II: Units and Fields

4. Study the proof of Proposition 4.41, making sure to look up references. Are there any steps that are

unclear?

Would a similar proof work for Z[!]? (You don’t need to work out the details now, but one of your

discussion problems for this section is to describe the units in Z[!]. It might help to reread

Proposition 3.38, on page 121.)


Math 301, 4.3 Commutative Rings-III: Subrings and Subfields

Name:

Read Section 4.3, pages 166-167, Subrings and Subfields. This is a short reading assignment, so take the

time to read it carefully.

Reading Questions

1. There are just two definitions in this reading assignment. What are the two terms that are defined?

(Write them here.) Write the definitions in your notebook, word for word.

2. Do you ever read the margin notes? There’s a good question (a “query”) in the margin near the first

definition. State and answer the question here.

3. Proposition 4.46 and Proposition 4.48 give necessary and su�cient conditions for a subset of a ring to

be a subring and for a subring of a field to be a subfield, respectively. (See the corrected version of

Proposition 4.46.) Write these propositions in your notebook, word for word.

Study the proof of Proposition 4.46 (posted online). Do you have any questions about this proof?

4. Summarize the paragraph after the proof of Proposition 4.46 in your own words. Why is the

proposition powerful?

1

Math 301, 4.3 Commutative Rings-III: Subrings and Subfields

5. Example 4.47 is about a Boolean ring. We will not study such rings in detail, but it is good to be

aware of them since they are di↵erent from many of the other rings we study. The paragraph after

the example explains this; summarize that paragraph here in your own words.


2

Math 301, 5.1 Domains and Fraction Fields

Name:

Reading Questions

1. (a) Write the definition of a domain in your notebook, word for word.

(b) Sometimes a more explicit version of the definition is useful. Finish the sentence:

A domain D is a nonzero commutative ring satisfying the following condition:

if a, b 2 D with a 6= 0, b 6= 0, then ab .

(c) The contrapositive of the above condition is sometimes more useful in proofs. Finish thesentence:

A domain D is a nonzero commutative ring satisfying the following condition:

if a, b 2 D with ab = 0, then either or .

(d) Proposition 5.1 gives another way to characterize a domain. Write the proposition in yournotebook, word for word, and read the proof carefully. Do all the steps of the proof make sense?

2. We have now defined three abstract structures: commutative rings, fields, and domains. How arethey related? For example, is every commutative ring a domain? (No. Give an example.) Is everyfield a domain? Is every domain a field? Cite definitions or propositions; give examples.

1

Math 301, 5.1 Domains and Fraction Fields

3. The first paragraph under Fraction Fields explains gives an overview and motivation for the discussionthat follows. Summarize this paragraph in your own words. (What are we going to do and why?)

4. State the definition of a fraction field. (In order to do this, you need to back track a bit, becausethere is notation in the definition that is not explained in the definition itself. In particular, whatdoes [a, b] represent? It’s an equivalence class, but what’s the equivalence relation?)


2

Math 301, 5.2 Polynomials

Name:

In this section, we give a precise and formal definition of a polynomial. It takes quite a bit of work to do

this! Read the notes on Section 5.2, referencing the text for the details, as needed, and take notes in your

notebook, as appropriate. Then answer the following reading questions.

Reading Questions

1. By plugging in all possible values of x in Z7, check that the two functions f(x) = x

7+ 2x� 1 and

g(x) = 3x+ 6, when considered as functions Z7 ! Z7, are actually the same function.

2. Use the definition of addition of formal power series to compute the following sums.

(a) (1, 1, 1, 1, 1, 1, . . . ) + (2, 0, 2, 0, 2, 0, . . . )

(b) (2, 0, 0, 0, . . . ) + (0, 1, 0, 0, . . . ) + (0, 0, 3, 0, 0, . . . )

3. Use the definition of multiplication of formal power series to compute the following products.

(a) (0, 0, 0, . . . ) · (1, 2, 3, 4, . . . )

(b) (1, 0, 0, 0, . . . ) · (1, 2, 3, 4, . . . )

1

Math 301, 5.2 Polynomials

(c) (5, 0, 0, 0, . . . ) · (1, 2, 3, 4, . . . )

(d) (0, 1, 0, 0, . . . ) · (0, 1, 0, 0, . . . )

(e) (0, 1, 0, 0, . . . ) · (0, 0, 1, 0, . . . )

4. Check that there are only four functions Z2 ! Z2. Write input-output tables for each one.


2

Math 301, 5.3-I Homomorphisms

Name:

Reading Questions

Read Section 5.3, pages 206-211. You may omit Example 5.15(v) and Example 5.16. There are onlytwo definitions and one proven result in this part of the reading, but there are a lot of good examples.Read them carefully, making sure to look up references, and work out the details for yourself!

1. What are the two terms that are defined? (Write them here.) Write their definitions, word for word,in your notebook.

2. Reread Example 5.14 and 5.15.

(a) One of the examples ends with a question. State and answer the question here.

(b) Are any of the examples unclear? Which ones? Take the time to reread them, look upreferences, and check the details in your notebook.

(c) Challenge: Can you think of any additional examples? I can think of at least two examples thatwe’ve seen; one was in the D 3.2-I assignment and the other in the D 4.3-III assignment.

1

Math 301, 5.3-I Homomorphisms

3. This section contains a result stating properties of homomorphisms. Write this result, word for word,in your notebook.

4. Example 5.18 states two claims about finite commutative rings, and explains why both claims aretrue. What are the two claims? Briefly summarize the arguments explaining why the claims are true.

Skim Section 5.3, pages 213-216. The main idea is that when we have a homomorphism of rings R ! S, (i)we can extend the homomorphism to a homomorphism R[x] ! S (Theorem 5.19) and (ii) to ahomomorphism R[x] ! S[x] (Corollary 5.22). These results have very useful applications.

5. Corollary 5.21 is a useful application of Theorem 5.19. Write the definition of the evaluationhomomorphism (given at the top of page 215) and Corollary 5.21 in your notebook, word for word.

What is e3(2� 3x+ x

2)?

6. Examples 5.23 and 5.24 are useful applications of Corollary 5.22.

(a) If r2 : Z ! Z2 is reduction modulo 2, what is r⇤2(3 + 2x+ 5x2)?

(b) If c : C ! C is complex conjugation, what is c⇤((2 + 4i) + (2 + 3i)x+ x

2)?


2

Math 301, 5.3-II Homomorphisms

Name:

Read Section 5.3, pages 216-220, Kernel, Image, and Ideals.

Reading Questions

1. Write the definitions of the kernel and image of a homomorphism in your notebook, word for word,

and study the examples given in Example 5.26. If any of these examples are unclear to you, take the

time to reread them and work out the details in your notebook.

2. Write the definition of an ideal in your notebook, word for word. Restate Proposition 5.25 as two

claims (instead of three) using the word ideal.

3. The author points out that we have seen ideals in this class outside the context of kernels. In

particular, one of your written problems turns out to have been about ideals in Z. What problem is

it? Restate this problem using the word ideal.

4. List (here) the additional terminology defined on page 218 and 219. Take notes in your notebook.

5. Simple examples of ideals are given in Example 5.27. Make sure you understand these. Pay

particular attention to 5.27(iv).

6. Example 5.30 contains a result worth remembering. Summarize the example here. (The power of this

result is illustrated in Corollary 5.32.)

1

Math 301, 5.3-II Homomorphisms

7. Write Proposition 5.31 in your notebook, word for word, and look at the proof. Take the time to fill

in the details of the proof for yourself, if there are any parts that are unclear to you.


2

Math 301, 6.1-I Parallels to Z

Name:

Read Section 6.1, pages 233-239, up to but not including Roots. You may omit Example 6.7.

Reading Questions

1. There are three terms that are defined for a general commutative ring. State the terms here, andwrite their definitions in your notebook, word for word.

2. With the exception of Proposition 6.6(i), all the results in this part of the section are aboutpolynomial rings. Most of the results are about k[x], where k is a field. The two main results areanalogues of Proposition 1.14 and Theorem 1.15, namely Proposition 6.8 and Theorem 6.11. Writeboth of these results in your notebook, word for word, and briefly summarize them here.

3. Lemma 6.1 is very useful and is used in the proofs of many of the other propositions in this section.Read through the proof again, making sure to look up any references. Would this proposition be trueif k were merely a domain and not necessarily a field?

4. Proposition 6.2 describes the units in k[x], where k is a field. This result is also used in the proofs ofmany results in this section. Reread the proof. Compare with Exercise 5.14.

Challenge: Would the proof still work if k were merely a domain and not necessarily a field?

1

Math 301, 6.1-I Parallels to Z

5. Reread Proposition 6.4 and its proof. Find a monic associate for 4x2 + 6x+ 11 in Q[x]. Does4x2 + 6x+ 11 have a monic associate in Z[x]? Explain.

6. Proposition 6.5 gives a useful criterion for irreducibility in k[x], where k is a field. Remember thatthe results about k[x] in this section are analogues of results about Z. Reread Proposition 6.5, andtry to formulate the analogous statement for Z.

Hint: Nonzero constants in k[x] are the units in k[x] and the units in Z are ±1, irreducibles in Z are±p for primes p, and the analogue of degree in k[x] is absolute value in Z.

7. What do you think the analogue of Proposition 6.6(ii) would be for Z? (Hint: The analogue of amonic polynomial is a positive integer.)

8. Proposition 6.13 will be useful for future discussions about roots of unity. Look back at Exercise 3.40on page 106. List all the polynomials of the form x

m � 1, for 0 < m 2 Z, that are divisors of x12 � 1.


2

Math 301, 6.1-II Parallels to Z

Name:

Read Section 6.1, Roots, GCDs, and the first page of Unique Factorization, pages 239-248. You may

omit Proposition 6.21.

Reading Questions

1. The first four results in the part about roots build on each other: the Remainder Theorem is used to

prove the Factor Theorem, the Factor Theorem is used to prove Theorem 6.16, and Theorem 6.16 is

used to prove Proposition 6.18. Take some notes on them. The Factor Theorem and Theorem 6.16

are generally useful; write these in your notebook word for word.

2. Give an example to show that Theorem 6.16 is not true for polynomials with coe�cients in an

arbitrary commutative ring.

3. Study the proof of Proposition 6.18. Would this proof work if k was a finite field? Prove or give a

counterexample.

4. Write the definition of a gcd of two polynomials in k[x] (where k is a field) in your notebook, word for

word. Why do we define a gcd to be monic?

5. Theorem 6.25 is the analogue of Theorem 5.29. The outline of the proof should be familiar to you by

now! Make sure you understand it. Rereading the proof of Theorem 5.29 may help.

1

Math 301, 6.1-II Parallels to Z

6. Reread Example 5.27(iv) and Corollary 6.26, with its proof. Suppose J is the ideal in Q[x] consisting

of all polynomials having

p2 as a root. Can you think of a polynomial d(x) 2 Q[x] such that J = (d)?

7. Theorem 6.28, Corollary 6.29, Theorem 6.31, Corollary 6.32, and Proposition 6.33 all have analogues

in Section 1.3. What are the analogues? (You don’t need to give the statement of the results, just the

names, e.g. “Theorem 1.X.”)


2

Math 301, 6.1-III Parallels to Z

Name:

Read Section 6.1, Unique Factorization and Principal Ideal Domains, pages 249-258.

Reading Questions

1. Pages 249-251 discuss the Euclidean Algorithm for polynomials in k[x], where k is a field. Recall that

the Euclidean Algorithm is basically a repeated use of the Division Algorithm. You may find it

helpful to review the Euclidean Algorithm for integers (p 32-33). Read through Examples 6.35, 6.36,

6.39 to see how the Euclidean Algorithm works for polynomials. Notice that the computations

quickly become cumbersome, and the authors use a computer to do the long division.

In Example 6.39, the gcd of f(x) = x

3 � 2x

2+ x� 2 and g(x) = x

4 � 1 is computed. What is it?

What are the coe�cients s and t such that gcd(f, g) = s f + t g?

2. Write Theorem 6.40 word for word in your notebook. The exact wording is important. Theorem 6.40,

Proposition 6.41, and Corollary 6.42 all have analogues in Section 2.1. What are the analogues? (You

don’t need to give the statement of the results, just the names, e.g. “Theorem 1.X.”)

3. The last paragraph before the exercises defines the multiplicity of a root. Construct an example for

yourself to make this definition more concrete. (Pick your favorite number r to be a root and your

favorite positive integer e to be the multiplicity of the root. Construct a polynomial that has r as a

root with multiplicity e.)

4. Write the definition of a PID in your notebook, word for word, and study Example 6.44, especially

(i)-(iv), making sure to look up references. For (iii), you might want to look back at Example 5.30(ii).

5. By definition, every PID is a domain. Is every domain a PID? Is every field a PID? Is every PID a

field? Support your claims with references to examples or proven results.

1

Math 301, 6.1-III Parallels to Z

6. Write the definition of a gcd in a PID in your notebook, word for word. Does this match the

definition given for a gcd in Z or k[x]?

7. In the next several pages, the author generalizes the discussion of unique factorization in Z, given in

Section 2.1, and unique factorization in k[x], given in the preceding parts of Section 6.1, to prove

unique factorization in an abstract PID.

8. Write the definition of a UFD in your notebook, word for word.

9. Reread the paragraphs after the definition of UFD. By definition, every UFD is a domain. Is every

domain a UFD? Is every PID a UFD? Is every UFD a PID? Support your claims with references to

examples or proven results.


2

Math 301, 6.2 Irreducibility

Name:

Reading Questions

1. This section contains several (five, at my count) irreducibility criteria (i.e. results of the form “If . . . ,then f(x) is irreducible in (some polynomial ring.)” Give a brief description of each one here, andwrite them, word for word, in your notebook.

2. One of these criteria is useful only for polynomials of low degree. (Which one?) Use this criterion,along with the Rational Root Theorem, to show that x2 + x+ 1 is irreducible in Q[x].

3. The Rational Root Theorem is useful because it allows us to reduce the list of possible roots from aninfinite list (every rational number) to a finite list (rational numbers of a certain form.) This is alsothe reason why reduction modulo p (for prime p) is useful. Recall that Fp is the (unique!) finite fieldwith p elements. Thus Fp is the same as Zp. (See page 205.)

Show that x2 + x+ 1 is irreducible in F2[x] by checking all the possible roots (namely [0] and [1].)What does this allow us to conclude about x2 + x+ 1 as a polynomial in Z[x]? Why?

Show that x4 + 2x3 + 3x+ 7 is irreducible in Z[x] using reduction mod 2.

1

Math 301, 6.2 Irreducibility

4. Write the definition of the d

th cyclotomic polynomial in your notebook, along with Proposition 6.60.

Use Proposition 6.60, along with the table on page 266, to write explicit factorizations for x� 1,x

2 � 1, x3 � 1, x4 � 1, x6 � 1 and x

12 � 1.


2

Math 301, A.1, A.2 Functions and Equivalence Relations

Name:

Reading Questions

1. Appendix 1 is a formal treatment of functions. It begins with motivation: the author is trying to

convince you that a formal notion of function will be useful. Reread the first page of the appendix,

and explain, in your own words, why the informal definition given in most calculus books is

problematic.

2. The next several pages contain a lot of formal definitions. Reread and pay particular attention to the

definitions of: function, equality of functions, well-definedness and single-valuedness, injection,

surjection, and bijection. Write these definitions (word for word) in your notebook. Which of these

definitions are confusing to you?

3. This appendix also contains examples to illustrate the definitions. Carefully reread the examples that

pertain to the definitions that were confusing to you. Take some notes (in your notebook) on the

examples that help you to understand the definitions. Give an example or two of your own below.

4. Appendix 2 discusses equivalence relations. It begins with a motivating example. What is the

motivating example?

Math 301, A.1, A.2 Functions and Equivalence Relations

5. The main ideas in this appendix are the definitions of equivalence relation, equivalence class, and

partition, and the proposition that relates equivalence relations to partitions. Reread these definitions

and the proposition, and write them in your notebook. Which of these ideas are confusing to you?

6. Some of the examples in this section pertain to topics we have not yet covered, namely Example A.14

(ii) and (iii) and A.15 (ii) and (iii). The other examples should be accessible to you from your

previous math experience. Carefully reread the examples, paying special attention to A.14 (iv) and

A.15 (iv). Take notes on the examples that are especially helpful to you in understanding the

definitions or the proposition. Can you think of your own examples?


Math 301, 7.1 Quotient Rings

Name:

Read Section 7.1, Quotient Rings, pages 277-284.

This section generalizes the construction of the commutative ring, Zm, from the ring of integers, Z. As theauthors develop the theory for a general ring, R and ideal I, they tell you where to find the correspondingresults for Zm. Be sure to look back at those results when directed to do so. Familiarity with Z and Zm

will help you in understanding the generalization.

Reading Questions

1. For any ideal I in a commutative ring R, “congruence mod I” can be used to build anothercommutative ring that generalizes the construction of Zm from Z. Describe the construction of thisquotient ring, R/I, in your notebook.

Describe the elements of R/I (what are these elements called?) and the addition and multiplicationin R/I.

Why are these operations well defined in R/I?

What is the zero of this ring?

What is the multiplicative identity in this ring?

2. For the ideal I = (x2 � 2) in Z[x], compute the product (x+ 3 + I)(x2 + 2x� 1 + I) in the quotientring Z[x]/I. Hint: Since “x2 � 2 = 0” in Z[x]/I, “x2 = 2” in Z[x]/I.

1

Math 301, 7.1 Quotient Rings

3. In Proposition 7.8, we define the natural map ⇡ : R ! R/I by a ! a+ I, and prove that this is asurjective homomorphism. Write the definition of this homomorphism in your notebook. Whatprecisely is this map in the case where R = Z and I = (m)? When/where have you proven directlythat this is a homomorphism?

4. Write the statement of Theorem 7.10, the First Isomorphism Theorem, in your notebook, word forword. What conclusion does this theorem allow you to make about the commutative ring Zm? Doesthis makes sense to you?

5. Theorem 7.11 tells us that if I = (x2 + 1), then R[x]/I is a field isomorphic to C. Compute theproduct (2x+ 3 + I)(3x� 4 + I) in R[x]/(x2 + 1).

Find the inverse of (2x+ 3 + I) in R[x]/(x2 + 1) (i.e. find a, b 2 R, such that(2x+ 3 + I)(ax+ b+ I) = 1 + I.)


2

Math 301, 7.2-I Field Theory: Characteristics and Extension Fields

Name:

Read Section 7.2 Field Theory, Characteristics and Extension Fields, pages 287-292. Reference AppendixA.3 on Vector Spaces, as needed.

Reading Questions

1. The main idea in the discussion on characteristics is contained in the first paragraph. What are thetwo types of fields?

2. (a) Consider the map � : Z ! R given by �(n) = 1 + 1 + · · ·+ 1 (n times). What is the kernel of �?

(b) Now consider � : Z ! F2 given by �(n) = [1] + [1] + · · ·+ [1] (n times). What is the kernel of �?

(c) What is the characteristic of R? Of F2?

3. Propositions 7.17 and 7.18 are both important and will be used later on. Write them word for wordin your notebook.

4. True or false. (If true, cite a result in the section; if false, provide a counterexample.)

(a) All infinite fields are of characteristic zero.

(b) There is no finite field with 24 elements.

1

Math 301, 7.2-I Field Theory: Characteristics and Extension Fields

5. The main result in the discussion of Extension Fields is Proposition 7.20.

The idea is that, given a “base field” k and a monic irreducible polynomial p(x) in k[x], we willconstruct an “extension field” K of k such that K contains a root of p(x).

We have already done this in the case where the base field is k = R and the irreducible polynomial isp(x) = x

2 + 1. See Example 5.27(iv), Corollary 6.26, and Theorem 7.11. The extension field isK = R[x]/(x2 + 1), which is isomorphic to C, under the isomorphism '

�a+ bx+ (x2 + 1)

�= a+ bi.

Even though R is not literally a subfield of R[x]/(x2 + 1), we consider it as a subfield because there isan isomorphic copy of R inside R[x]/(x2 + 1), namely {a+ (x2 + 1) : a 2 R}. This corresponds to{a+ 0i : a 2 R} ⇢ C, which is also a copy of R.

The “root” of x2 + 1 in R[x]/(x2 + 1) is the coset x+ (x2 + 1), which corresponds to i, under theisomorphism ', which is obviously a root of x2 + 1 in C.

In Proposition 7.20, the extension field K is also constructed as a quotient, k[x]/(p). This quotientcontains a copy of the base field; the copy is called k

0 in the proposition. The “root” of p(x) in theextension field is the coset x+ (p), which is denoted as z, for brevity.

State parts (iii), (iv), and (v) of Proposition 7.20 in the specific case k = R, p(x) = x

2 + 1.

6. Work carefully through Example 7.22 to help you understand what Proposition 7.20 is saying. Wewill continue working with this example throughout Chapter 7. (It will be useful later on to noticethat the polynomial p(x) in this example is exactly the polynomial you found in Exercise 3.56.)

7. Write Proposition 7.19 and 7.20, Corollary 7.21 as well as the definitions of extension field, finiteextension, and the degree of a field extension in your note book, word for word.

If k = R and K = R[x]/(x2 + 1), what is the degree of K/k?


2

Math 301, 7.2-II Field Theory: Algebraic Extensions

Name:

Read Section 7.2 Field Theory, Algebraic Extensions, pages 293-299. Reference Appendix A.3 on VectorSpaces, as needed. Also see the hand-out containing additional examples.

Reading Questions

1. This section contains many important defintions. In particular, make sure you write the definitions ofthe following terms in your notebook, word for word: algebraic element (what does it mean for anelement in a field extension to be algebraic over the base field), algebraic field extension,adjoining (what does it mean to adjoin an element or set of elements of a field extension to the basefield), minimal polynomial.

2. (a) Explain why z = 3p2 is algebraic over Q. (Find a nonzero polynomial with rational coe�cients

that has z as a root. There are many possible answers for this.)

(b) Find the minimal polynomial p(x) for z = 3p2 over Q. (Find a monic irreducible polynomial

p(x) in Q[x] that has z as a root. There is only one correct answer for this.)

(c) By Theorem 7.25, Q[x]/(p) ⇠= Q(z) = Q( 3p2). What is the dimension of Q( 3

p2) as a vector space

over Q? (Use Proposition 7.20(iv) and (v).) What is the degree of Q( 3p2) over Q? (Use

Corollary 7.21.)

(d) Find the two other roots, z0 and z

00, of p(x). (Note: if you remember what we learned aboutcube roots in Chapter 3, you should not need to do any computation here.) What does Theorem7.25(ii) say about the relationship between Q(z), Q(z0) and Q(z00)?

1


3. Consider the homomorphism ' : Q[x] ! R given by '(f) = f( 3p2).

(a) What is the kernel of ' in Q[x]? (Be explicit: “The kernel of ' consists of all polynomials inQ[x] that . . . ”)

(b) Since the kernel is a nonzero ideal in Q[x], Theorem 6.25 ensures that the kernel is generated bya unique monic polynomial in Q[x]. What is this polynomial?

(c) What is the image of ' in R? (Again, be explicit: “The image of ' consists of all real numbersof the form . . . ”)

(d) What does the First Isomorphism Theorem allow you to conclude in this situation? (Use youranswers in (a) and (b) to be explicit.)

(e) The image of ' is a subring of R. What does Proposition 5.3 imply about the image of '? Whatdoes this allow us to conclude about the quotient ring in (c)? (Recall Exercise 5.29(ii).)

(f) What does Proposition 7.19 imply about the polynomial generating the kernel of '?

2


4. Example 7.28 illustrates the idea of a tower of field extensions. Reread it carefully, and take notes onit. Looking back at Exercise 3.56 and Example 7.22 may help.

(a) Let ⇣ = cos(2⇡/7) + i sin(2⇡/7). What is the minimal polynomial for ⇣ over Q? How do weknow it is irreducible over Q?

(b) What is the degree of Q(⇣)/Q?

(c) Let ↵, �, and � be as in the example. What is the minimal polynomial for ↵ over Q? What isthe degree of Q(↵)/Q?

(d) How do we know Q(⇣) contains ↵? Note that this implies Q(↵) ⇢ Q(⇣), since Q(↵) is, bydefinition, the intersection of all subfields of C containing ↵.

(e) What is the degree of Q(⇣)/Q(↵)? (Use Theorem 7.27.)

(f) What is the minimal polynomial of ⇣ over Q(↵)?

(g) With � and ✏ as defined in the example, show that � + ✏ = �1 and �✏ = 2. (Use Theorem 3.32.)

3


(h) What is the minimal polynomial of � over Q? What is the degree of Q(�)/Q?

(i) How do we know Q(�) ⇢ Q(⇣)? What is the degree of Q(⇣)/Q(�)?

(j) What is the minimal polynomial of ⇣ over Q(�)? (Look back at Exercise 3.15 to see why thecomputations done at the end of the example do yield a polynomial having ⇣ as a root.)


4

Math 301, 7.2-III Field Theory: Splitting Fields, Classification of Finite Fields

Name:

Read Section 7.2 Field Theory, Splitting Fields and Classification of Finite Fields, pages 300-307.

Reading Questions

1. In the discussion of splitting fields, pay particular attention to Kronecker’s Theorem (Theorem 7.29),the definition of splitting field, and the results showing the existence (Corollary 7.30) anduniqueness-up-to-isomorphism (Corollary 7.36) of splitting fields.

2. (a) Write Kronecker’s Theorem in your notebook, word for word.

(b) For f(x) = x

2 + 1 2 Q[x], give an example of a field extension K/Q such that f is a product oflinear factors in K[x]. (There are several possible answers.)

(c) For f(x) = x

2 � 2 2 Q[x], give an example of a field extension K/Q such that f is a product oflinear factors in K[x]. (There are several possible answers.)

3. (a) The definition on page 301 states what it means for a polynomial to split over a field extensionand what it means for a field extension to be a splitting field of a polynomial. Write thisdefinition in your notebook, word for word.

(b) What is the splitting field of x2 +1 over Q? What is the splitting field of x2 +1 over R? Explainwhy they are di↵erent.

(c) What is the splitting field of x2 � 2 over Q? Over R? Explain.

1

Math 301, 7.2-III Field Theory: Splitting Fields, Classification of Finite Fields

4. The first paragraph in the discussion of classification of finite fields (page 305) sums up what wealready know about finite fields (two results) and what we will prove about finite fields in this section(also two results). State these four results below.

5. Study the proof of Galois’ Theorem (Theorem 7.38), making sure to look up references. Here is theoutline. Given a prime p and a positive integer n, we will construct a field E with q = p

n elements.We will define E be a set of roots of a certain polynomial g(x) 2 k[x]. Two things must be shownabout E: that it has exactly q elements and that it is a field.

(a) Describe the set-up needed to define E as a set. In your description make sure to answerfollowing questions. What is the polynomial g(x) 2 k[x]? (What is g and what is k?) What isthe field extension K containing the roots of g? (What theorem do we need to use?)

(b) How do we show that E has exactly q elements?

(c) What do we need to show in order to prove that E is a field?


2

Math 301, 8.1 Arithmetic in Gaussian and Eisenstein Integers

Name:

Read the introduction to Chapter 8 (pages 329-330) and Section 8.1 (pages 330-336). It may help to review

the definition and properties of the norm (pages 116-117) in Section 3.4 and units and associates in Z[i]and Z[!] (Example 6.3(i), (ii), page 234).

Reading Questions

1. The introduction to Chapter 8 gives the motivation for studying arithmetic in rings of cyclotomic

integers and gives an overview of Chapter 8. Briefly summarize the motivation in your own words.

Section 8.1 describes the Division Algorithm and the Euclidean Algorithm in Z[i] and Z[!] and describes a

more general structure (called a Euclidean Domain) that captures the properties shared by Z, k[x] (where k

is a field), Z[i], and Z[!] which are necessary for the Division Algorithm and Euclidean Algorithm to work.

2. Reread Examples 8.1 and 8.3 carefully, working out the computations for yourself by hand on scratch

paper. Understanding these examples will shed light on the results in the rest of the section.

(a) Illustrate Example 8.1 by plotting the Gaussian integers in the complex plane, along with w/z

and q.

(b) Illustrate Example 8.3 by plotting the Gaussian integers in the complex plane, along with w/z

and all four choices of q.

(c) Reread the proof of Proposition 8.2 with these examples in mind.

1

Math 301, 8.1 Arithmetic in Gaussian and Eisenstein Integers

3. Illustrate Example 8.5 by plotting the Eisenstein integers in the complex plane, along with w/z and

q. Reread the proof of Proposition 8.4 with this example in mind.

4. Explain the relationships between Euclidean domains, PIDs and UFDs. (Is every Euclidean domain a

PID? Is every PID a Euclidean domain? Etc.) Give examples or citations to justify each statement.

5. Reread Example 8.9. What is the gcd of 91 + 84! and 34 + 53! in Z[!] that is computed in this

example? What are the associates of this gcd in Z[!]?


2

Math 301, 8.2-I Primes Upstairs and Primes Downstairs: Gaussian Primes

Name:

Read Section 8.2, pages 337-343.

Reading Questions

1. Study Example 8.12. A careful reading of this example will help the results in this section make moresense. What is the prime factorization of �211 + 102i in Z[i]? How do we know that the factors areprime in Z[i]? (Cite a proposition.)

2. There are many new terms in this section. Make sure to take notes on them. In particular, make sureyou understand what is meant by primes upstairs and primes downstairs, rational prime,rational integer, and a Gaussian prime lying above a rational prime.

3. A prime p 2 Z may factor in Z[i] or it may remain prime in Z[i]. Give an example of each.

4. Let p be a rational prime. State four conditions equivalent to “p factors in Z[i].”

1

Math 301, 8.2-I Primes Upstairs and Primes Downstairs: Gaussian Primes

5. A rational prime exhibits one of the following three behaviors in Z[i]: it splits, it is inert, or itramifies. In your notes, make sure to describe what each of these terms means. Give an example ofeach type of behavior here.

6. The main results in this reading are Theorem 8.21 (Law of Decomposition of Gaussian Integers) andCorollary 8.22 (Classification of Gaussian Primes.) Write these in your notebook, word for word.

7. The primes in Z[i] are of three types (as described in Corollary 8.22). Give an example of each type.


2

Math 301, 8.2-II Primes Upstairs and Primes Downstairs: Eisenstein Primes

Name:

Read Section 8.2, pages 344-348.

Reading Questions

1. Let p be a rational prime. State three conditions equivalent to “p factors in Z[!].”

2. A rational prime either splits, is inert, or ramifies in Z[!]. Give an example of each.

3. The main results in this reading are Theorem 8.29 (Law of Decomposition of Eisenstein Integers) and

Corollary 8.30 (Classification of Eisenstein Primes.) Write these in your notebook, word for word.

4. The primes in Z[!] are of three types (as described in Corollary 8.30). Give an example of each type.

1

Math 301, 8.2-II Primes Upstairs and Primes Downstairs: Eisenstein Primes


2

Math 301, 8.3 Fermat’s Last Theorem for Exponent 3

Name:

Read Section 8.3. Read for the big picture, and be careful not to get embroiled in the details!

Reading Questions

1. Reread the “How to Think About It” box at the bottom of page 349, and summarize it.

2. There are three (or four) important preliminaries (a proposition, a lemma, and a definition, with aproposition.) What are they?

3. If z = 1 + 16! = (�1)(5� !)(1� !)2, what is ⌫(z)?

4. What is “the first case”? What is “the second case”?

1

Math 301, 8.3 Fermat’s Last Theorem for Exponent 3

5. Gauss’ proof of the second case follows from a more general theorem. State this theorem.


2

Math 301, 8.4 Approaches to the General Case

Name:

Read Section 8.4. Focus on trying to understand the main ideas and how they fit together, and be careful

not to get bogged down in the details.

Reading Questions

1. Reread the first four paragraphs of the section. These provide an overview of the section. Summarize

the overview here, in your own words.

2. After recalling some facts about cyclotomic extensions at the bottom of page 360, we proceed to

investigate units in Z[⇣p], where ⇣p = cos(2⇡/p) + i sin(2⇡/p). We start by looking at how the prime pramifies in Z[⇣p].

(a) Write out the statement of Lemma 8.46 in the case p = 3 and in the case p = 5.

(b) Write out the statement of Proposition 8.47 in the case p = 3 and in the case p = 5. Note that

the condition that p - st could be replaced by the condition 0 < s, t < p. Thus for the case p = 3,

you need to consider s, t 2 {1, 2}, and for p = 5 you need to consider s, t 2 {1, 2, 3, 4}.

(c) Corollary 8.48 says that p ramifies in Z[⇣p]. For p = 3, when ⇣p = !, we have, “In Z[!] there is a

unit u such that 3 = u(1� !)2.” What is the unit u in this case? (Look back at 8.2.)

For p = 5, we have, “In Z[⇣5], there is a unit u such that 5 = u(1� ⇣5)4.”

1

Math 301, 8.4 Approaches to the General Case

(d) Corollary 8.49 gives us a way to generate units in Z[⇣p]. Since 1� ⇣sp and 1� ⇣tp are associates if

0 < s, t < p, dividing yields a unit.

For example, in Z[!], this gives us three units:

(1� !)/(1� !) = (1� !2)/(1� !2

) = 1

(1� !)/(1� !2) = �!

(1� !2)/(1� !) = �!2

(e) Proposition 8.50 gives even more units in Z[⇣p]. What unit does this give in Z[!]? Is it on the

unit circle in C? What unit does it give in Z[⇣5]? Is it on the unit circle?

3. In the second part of Section 8.4, we discuss the failure of unique factorization and how Kummer

dealt with it using “ideal numbers” (which are really ideals!) Since the smallest ring of cyclotomic

integers in which unique factorization fails is too big for hand computation (Z[⇣23]), we work in

another ring in which unique factorization fails, Z[p�5].

(a) Give an example that unique factorization fails in Z[p�5], and cite a specific result in the

section to back up your example.

(b) Skim pages 366-367, which discuss Kummer’s attempt to “repair” the non-uniqueness of

factorization using “ghost factors.” You will work further with Example 8.52 in the discussion

problems for this section.


2

Math 301, 9.4 Group Theory

Name:

Read the introduction to Chapter 9 (page 379) and the beginning of Section 9.4 (pages 389-390). Thebackground for Example 9.7 can be found in Appendix A.1 (page 418) and the background for Example 9.8can be found in Section 9.3 (page 386).

Reading Questions

1. What classical problem did Galois resolve by inventing group theory?

2. Write the following definitions in your notebook, word for word: group, subgroup, abelian group.

3. Give an example of a nonabelian group. (There is an example given in this reading.)

4. (a) Show that R with the binary operation of addition is an abelian group. (Note: Here the identityelement is e = 0, and the inverse of a real number a is �a.)

(b) Show that Z is a subgroup of R with addition.

5. (a) Show that the set of positive real numbers is an abelian group with multiplication. (Note: Herethe identity element is e = 1, and the inverse of a positive real number a is 1

a .)

1

Math 301, 9.4 Group Theory

(b) Is the set of positive integers a subgroup of the set of positive real numbers with multiplication?Why or why not?

6. Let R be a commutative ring.

(a) Show that R is an abelian group under addition.

(b) Let U be the set of units in R. Show that U is an abelian group under multiplication.


2

Math 301, 9.5 Wiles and Fermat’s Last Theorem

Name:

Read Section 9.5, skimming the pages 396-403 and focusing on pages 404-407.

Reading Questions

1. Wiles proved Fermat’s Last Theorem as a distant corollary of a specific case of a conjecture (theTaniyama-Shimura-Weil Conjecture) which says, roughly, that every elliptic curve is modular.One might reasonably suppose that an elliptic curve is an ellipse, but it is not! Most of this section isdevoted to explaining what an elliptic curve is and how it is related to an ellipse. Briefly:

(a) One may describe the arclength along an ellipse with an integral, as on page 397. (Do youremember using integrals to find arclength in calculus?)

(b) Integrals of a similar form to this one are called elliptic integrals. They arise in many practicalapplications including the compression of an elastic rod and the oscillation of a simple pendulum.

(c) By analogy with sine, which is the inverse function for

arcsin(x) =

Zx

0

1p1� t

2dt

Gauss studied inverse functions of elliptic integrals, which are called elliptic functions.

(d) We can think of the unit circle as the curve in R2 whose coordinates are given by (sin ✓, cos ✓).By analogy, Gauss, Abel and Jacobi considered elliptic curves, curves whose coordinates aregiven by elliptic functions.

2. (a) What is the precise definition of an elliptic curve?

(b) Give an example of a modern-day application of elliptic curves over Fq

.

3. (a) The definition of a Diophantine equation is given at the bottom of page 404. A simpleexample of a Diophantine equation would be x

2 + y

2 � 1 = 0. This is the equation of the unitcircle in R2. Recall that, in Chapter 1, we saw that rational points on the unit circle correspondto Pythagorean triples.

(b) Pages 400-403 describe how congruent numbers correspond to rational points on cubic curves ofthe form y

2 = x

3 � n

2x with y 6= 0, which correspond to the Diophantine equations

x

3 � n

2x� y

2 = 0.

(c) What is the Diophantine equation that arises from Fermat’s Last Theorem?

1

Math 301, 9.5 Wiles and Fermat’s Last Theorem

4. Page 407 gives Andrew Granville’s account of the recent history leading up to the proof of Fermat’sLast Theorem. Give a brief summary in your own words here.


2

math 301, 1.1 ancient mathematics name: reading...

Documents