DATE: Summer Term, 1996

INSTRUCTOR: John Beachy, Watson 355, 753-6753

OFFICE HOURS: MTWR 11:00-12:00, or by appointment.

TEXTS:
**Abstract Algebra,** *2nd Ed.,* Beachy/Blair
**Elements of Abstract Algebra,** Clark

SYLLABUS: (Chapters in **Abstract Algebra,** *2nd Ed.*)

Review of basic group theory (Chapter 3, sections 3.6-3.8)

Structure of groups (Chapter 7)

Galois theory (Chapter 8)

Supplementary notes on groups

COURSE OBJECTIVES: The primary focus is on Galois theory, and the necessary group theory prerequisites. This elegant theory provides a complete answer to the question of when an equation can be solved by radicals.

A major goal is to provide preparation for the comprehensive exams.
These are the required topics:

**Groups:**
Groups, subgroups, normal subgroups, homomorphism theorems,
Sylow theorems, structure theorem for finite abelian groups,
Jordan-Holder theorem, solvable groups.

**Fields and Galois theory:**
characteristic, prime fields,
algebraic and transcendental extensions,
separability, perfect fields, normality, splitting fields,
Galois group, fundamental theorem of Galois theory,
solvability by radicals, structure of finite fields.

**Rings:** (covered in MATH 421)
Rings, ideals, homomorphisms, field of
fractions of an integral domain.

**Linear algebra:** (covered in MATH 423)
Linear independence, basis, dimension, direct sums,
linear transformations and their matrix representations,
linear functionals, dual spaces,
determinants, rank, eigenvalues and eigenvectors,
minimal and characteristic polynomials, canonical forms.

COURSE PREREQUISITE: MATH 421

GRADING: Final grades will be based on 400 points: 2 examinations (300) and homework (100).

I encourage studying in groups, and you may discuss homework problems with other students. You should write up your own solutions--direct copying is unacceptable. As a rough guideline for writing up solutions to homework problems, you should include enough detail so that (i) you can convince me that you understand the solution and (ii) you can understand your solution when you study for the final exam.

TENTATIVE SCHEDULE OF LECTURES:

Week Sections 1 Review 3.6-3.8 Permutation groups, Homomorphisms, Factor groups 5.1-5.4 Commutative rings 2 7.1-7.3 Isomorphism theorems; Conjugacy; Group actions 3 7.4-7.5 Sylow theorems; finite abelian groups 4 7.6-7.8 Solvable groups; simplicity of the alternating group EXAM I on Thursday, July 11 5 6.4-6.5 Splitting fields; finite fields 6 8.1-8.3 Fundamental theorem of Galois theory 7 8.4,8.6 Solvability by radicals; computations 8 notes Semidirect products; groups of small order EXAM II on Wednesday, August 7 HOMEWORK ASSIGNMENTS Due Section Page Problems 6/25 3.8 156 13, 14 #3 Describe the quaternion group via generators and relations; show that every subgroup is normal #4 The dihedral group of order 8 and the quaternion group are not isomorphic; show that a certain matrix group is isomorphic to the dihedral group 7/2 7.1 279 12, 13 7.2 285 4, 5, 7, 8, 10, 11, 14 7/9 7.3 290 1, 2, 8 7.4 295 6, 7, 8, 10, 11 7/18 7.6 307 5, 6 7.7 314 12 7/25 6.4 254 1c, 3c, 8, 9, 11, 12 7/31 8.1 321 2, 3, 4, 5 8/6 8.3 334 1, 3 8.4 339 1, 2

REVIEW FOR EXAM I (on 7/11/96)

Review some of the basic results: Cayley's theorem (3.6.2); dihedral groups (3.6.3); kernels (3.7.4); subgroups of factor groups (3.8.6); normal subgroups (3.8.7); the fundamental homomorphism theorem (3.8.8).

Section 7.1: Be sure you know the proofs of 7.1.1, 7.1.2, 7.1.3. This section provides some basic tools.

Section 7.2: The most important tool in this section is the conjugacy class equation (7.2.6). You need to know all of the results and examples, and the proofs of 7.2.8 (Burnside's theorem), 7.2.9., and 7.2.10 (Cauchy's theorem).

Section 7.3. In this section there are two tools: the generalized class equation given by any group action (7.3.6), and the generalization of Cayley's theorem relating group actions and homoorphisms into groups of permutations (7.3.2). You may omit the general version of Cauchy's theorem given in 7.3.8.

Section 7.4. As you know, the Sylow theorems are very important, and the previous sections have provided the necessary tools to prove them. Know the proofs of 7.4.1 and 7.4.4. You may omit 7.4.6. You need to know how to analyze the number of Sylow p-subgroups of a group, and how to apply this information.

Section 7.5. Know the proof of 7.5.1 (that a finite abelian group decomposes into the direct product of its Sylow subgroups). Know the statement of the fundamental theorem (7.5.4) and how to apply it. I won't ask you for a proof of the fundamental theorem or the lemmas leading up to it. I will not ask you about (7.5.7 - 7.5.11), although these results are very useful in constructing examples.

Section 7.6. The proof of 7.6.3 is important. You may omit the proof of the Jordan-Holder theorem.

Section 7.7. This section will not be covered on the test. If there is time before the test, I will prove that the alternating group on n elements is simple.

EXAM I, July 11, 1996

1. (40 pts)

(a) State the Sylow theorems.

(b) Carefully define the terms in the class equation
of a finite group, and use the class equation
to prove the first Sylow theorem
(on the existence of p-subgroups.)
You may assume Cauchy's Theorem.

2. (30 pts)

(a) Write out the definition of a solvable group.

(b) Prove that any p-group is solvable, where p is a prime number.

N.B. To give a complete proof you must prove
one of the following two facts:

A. The center of any finite p-group is nontrivial.

B. In any finite p-group, any subgroup of index p is normal.

3. (25 pts)

Assume that G is a group with |G|=p^{m}q^{n},
where p and q are distinct primes.
Assume further that there is only one Sylow p-subgroup H_{p}
and only one Sylow q-subgroup H_{q}.
Prove that G is isomorphic to the direct product
of H_{p} and H_{q}.

4. (35 pts) Groups of order 20

(a) List (up to isomorphism) all possible abelian groups of order 20.

(b) In a nonabelian group of order 20,
what are the possibilities for numbers of Sylow subgroups?

(c) For either the dihedral group of order 20
the Frobenius group of order 20,
find all Sylow subgroups and show explicitly
how they are conjugate to one another.

5. (20 pts)
Solve either part A or part B.

A.
Suppose G is a finite p-group, where p is prime.
Determine the conditions under which G is a simple group.

B.
In a group G, any element of the form xyx^{-1}y^{-1},
with x,y in G, is called a commutator of G.

(a) Find all commutators in the dihedral group of order 2n.

(b) Show that these commutators form a normal subgroup,
and that the corresponding factor group is abelian.

EXAM II, August 7, 1996

1. (50 pts)

(a) Find the Galois group of x^{3}-2 over the field
**Q** of rational numbers.

(b) Find the Galois group of x^{3}-2 over the field GF(5).

(c) Find the Galois group of x^{3}-2 over the field GF(7).

2. (30 pts)

(a) State the Fundamental Theorem of Galois Theory.

(b) Let F be any field, let G=Aut(F), and let H be a subgroup of G.
Define F^{H}, and prove that it is a subfield of F.
Prove that H is a subset of the set of elements of G that
fix all elements of F^{H}.
Prove that if K is a subgroup that contains H,
then F^{H} is a subset of F^{K}.

3. (15 pts)

Let F be a finite field with m=p^{n} elements,
where p is a prime number.
Prove that F is the splitting field of the polynomial
x^{m}-x over the prime subfield of F.

4. (20 pts)

Let K be a finite field, and let F be an extension of K with [F:K]=m.
Prove that Gal(F/K) is a cyclic group of order m,
and give an explicit formula for the generator of the group.

5. (15 pts)

Let K be a field, let f(x) belong to K[x],
and let F be the splitting field of f(x) over K.
Prove that if Gal(F/K) is cyclic,
then for each divisor d of [F:K] there is
exactly one field E between K and F with [E:K]=d.

6. (20 pts)

(a) Give the definition of an algebraic extension field.

(b) Prove that if E is an algebraic extension of K,
and F is an algebraic extension of E,
then F is an algebraic extension of K.

Solution to problem 1:

(a) The splitting field of x^{3}-2 over **Q**
is obtained by first adjoining the cube root of 2,
which has degree 3,
and then adjoining a primitive cube root of unity,
which has degree 2 over the previous extension.
(The primitive cube root of unity is not a real number,
so adjoining it does produce a proper extension.)
Thus the Galois group has order 6,
so since it is isomorphic to a subgroup of the symmetric group on 3 elements,
it must be the symmetric group S_{3}.

(b) Over GF(5), the irreducible factors of x^{3}-2
are x+2 and x^{2}+3x+4,
so the splitting field for x^{3}-2 has degree 2 over GF(5).
It follows that the Galois group is the cyclic group Z_{2}.

(c) Over GF(7), the polynomial x^{3}-2 is irreducible,
so its splitting field has degree 3 over GF(7).
It follows that the Galois group is the cyclic group Z_{3}.

Note that the Galois groups over GF(5) and GF(7) are isomorphic
to subgroups of the Galois group over **Q**,
as guaranteed by the general theory.

Solution to problem 5:

Let [F:K]=n.
We can use the Fundamental Theorem of Galois Theory
to translate the question about subfields into one about subgroups.
Recall that subgroups of a cyclic group of finite order
are in one-to-one correspondence with the divisors of the order of the group.
Thus we only need to note that if d is a divisor of n,
then n/d is a divisor of |Gal(F/K)|=n,
and so Gal(F/K) contains a subgroup of order n/d.
There is exactly one corresponding subfield E of F with [F:E]=n/d,
and so E is the unique subfield with [E:K]=d.

The remaining questions ask for proofs from the text.

RECOMMENDED TEXTS:

**Basic Algebra I, II,**
by Jacobson

An excellent book by a very good ring theorist.
It is my personal choice for a comprehensive reference book,
as it is well-written and contains a lot of material beyond 520 and 521.

**Topics in Algebra,**
by Herstein

A classic, ostensibly written for undergraduates.
Herstein's distinctive point of view makes it interesting reading.
The problem sets are superb (some are very difficult).

**Algebra, Vol I, II,**
by van der Waerden

Based in part on lectures in the 1920's by two giants in the field,
Emmy Noether and Emil Artin,
it shows its age even though it has been updated.
It is still an important reference,
and contains much that is of historical interest.

**Algebra, A Graduate Course,**
by Isaacs

This is a new book by a group theorist at UW-Madison.
Some texts are more comprehensive, but the topics chosen by Isaacs
do provide a coherent and interesting approach to algebra.
I recommend it highly.

**The Theory of Groups,**
by Rotman

One of the best introductory books on group theory.

**A Course in Galois Theory,**
by Garling

This paperback is interesting because it concentrates on Galois theory.

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