742

Math 742

Commutative Algebra and Galois Theory

Prof: Andrei Caldararu. Office hours: Wednesday 1:30pm, room VV 605.

Grader: Márton Hablicsek. Office hours: Tuesday 12:30pm, room VV 522, 263-7939.

Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.

This course, the second semester of the introductory graduate sequence in algebra, will cover the basic aspects of commutative ring theory and Galois theory. The textbook we'll use for the Commutative Algebra portion will be Atiyah-Macdonald "Commutative Algebra". For Galois Theory I plan on using "Fields and Galois Theory" by J.S. Milne which can be found here.

For Galois theory you may also look at Emil Artin's notes which are available here.

SYLLABUS

In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course.

WEEKS 0.5-1:

Category theory: the notions of category, functor, natural transformation. Full, faithful, essentially surjective functors. Examples. Groupoid. Categories enhanced in (abelian groups, vector spaces, topological spaces). Monomorphism, epimorphism.

WEEKS 1-2:

Commutative rings and homomorphisms, integral domains, fields. Ideals, prime and maximal. Existence of maximal ideals. Local rings. Some geometric pictures (Spec and Specm). Nilradical. The nilradical is the intersection of all primes. Relatively prime ideals, Chinese remainder theorem. Extension and contraction, pictures of what happens for the inclusion Z -> Z[i]

WEEK 3:

Modules, module homomorphisms. Module Hom, covariance and contravariance. Finitely generated modules, Nakayama's lemma and variants. Short exact sequences. Tensor product and exactness properties.

WEEK 4:

An A-B-bimodule M induces a functor Mod-A -> Mod-B given by -- \otimes_A M. The equivalence of categories Mod-k and Mod-M_n(k). Restriction and extension of scalars. Algebras. Tensor product of algebras, the case of C \otimes_R C. Exactness of tensor products, flatness.

WEEK 5

Simple rings. Structure of finite dimensional simple rings over a field. Brauer group, computation of Br(R) = Z/2Z. Semisimple modules and rings. Wedderburn and Artin-Wedderburn theorems. Maschke's theorem on semisimplicity of k[G] for a finite ring. Tensor product of algebras and interpretation as fiber product of affine schemes. Localization (definition and basic properties).

WEEK 6

Rings and modules of fractions. Definitions and universal properties. Examples A_f, A_p. Modules of fractions. The operation S^{-1} is exact, corresponds to tensoring with S^{-1}A, so S^{-1}A is flat. Local properties, examples. Ideals in rings of fractions, in particular primes in A_p are primes in A which are contained in p.

WEEK 7

Integral dependence and valuations. Various characterizations of integral dependence, integral closure forms a ring, which is integrally closed. Relations to rings of fractions. Going up and going down. Valuation rings definition and basic properties. Existence. The integral closure of A is the intersection of all valuation rings which contain A. Nullstellensatz.

Below you will find a repository of homework problems.

HOMEWORK 0 (not to be turned in)

1) Show that a fully faithful functor F: C -> D captures the property we called "essentially injective": if F(A) is isomorphic to F(B) for objects A, B of C, then A is isomorphic to B. (If you wanted to think of "functors which are injective on objects", and replaced equality with isomorphism, you'd get this notion of "essentially injective".)

2) Prove that a natural transformation eta: F => G (where F, G are functors C -> D) such that eta_A is an isomorphism for every A in C, is a natural isomorphism. (A natural transformation is called a natural isomorphism if there exists another natural transformation mu: G => F such that eta o mu = id_G, mu o eta = id_F.) In this case we say that F and G are naturally isomorphic.

3) Prove that a functor F that is fully faithful and essentially surjective is an equivalence, in the sense that there exists a functor G: D -> C and natural isomorphisms between F o G and id_D, and between G o F and id_C. (You will need to use the axiom of choice.)

4) Consider the functor ** : Vect -> Vect which takes a vector space V to its double dual V^**. Show that it is isomorphic to the identity when restricted to the subcategory of finite dimensional vector spaces. (We already constructed a map of functors id => ** in class.) If V is not finite dimensional, can you characterize the image of the natural map V -> V^**?

5) Fill in the blanks: "A category C, enhanced in abelian groups, with only one object, is the same things as a ...........". Same question, with "abelian groups" replaced by "k-vector spaces" for a fixed field k.

HOMEWORK 1 (due Feb 7)

Atiyah-Macdonald, page 10: 1, 2, 6, 10, 12, 15, 16, 17, 18, 21

HOMEWORK 2 (due Feb 14)

Atiyah-Macdonald, page 10: 19, 22, 26, 27, 28.

HOMEWORK 3 (due Mar 5)

Atiyah-Macdonald, page 31: 2, 3, 4, 8, 9, 10, 11 (second part is hard!), 12, 13, 14, 20. Bonus -- if you know about Tor -- do #24.

HOMEWORK 4 (due Mar 14)

Some non-commutative algebra exercises. You may find some references to read here. Or, even better, you can go and read Chapter I of the excellent book "Noncommutative Algebra" by Benson Farb and R. Keith Dennis. (On reserve in the library.) Most of these exercises are from this book.

1) Let A, A' be k-algebras, with subalgebras B, B', respectively. If the centralizers of B, B' are C, C' (in A, A', resp.), then the centralizer of B\otimes_k B' in A\otimes_k A' is C\otimes C'.

2) Find all the two-sided ideals of M_n(R), where R is a ring. Conclude that if D is a division algebra, M_n(D) is simple. (Hint: all two-sided ideals are of the form M_n(I) for some two-sided ideal I of R.)

3) Let N be a submodule of the R-module M. If N and M/N are semisimple, does it follow that M is semisimple?

4) Let M be a module such that every submodule is a direct summand. Show that M is semisimple as follows:

(a) Show that every submodule of M inherits the property that every submodule is a direct summand.

(b) Show that M contains a simple submodule: choose any finitely generated non-zero submodule M' of M. Let M" be a maximal submodule of M' such that M" is not equal to M' (why does it exist?). Then M'/M" is simple.

(c) Let M_1 be the submodule of M generated by all simple submodules. Show that M_1 = M.

5) Let A be a simple k-algebra with center k such that [A:k] = p^2 for a prime number p. Prove that A is either a division algebra or A is M_p(k).

HOMEWORK 5 (due Mar 28)

Atiyah-Macdonald page 31: 5, 6, 10; page 43: 1, 3, 5, 12, 13, 14

HOMEWORK 6 (due Apr 4)

Atiyah-Macdonald page 43: 15, 18, 19, 20, 21, 22, 25; page 67: 3, 9

HOMEWORK 7 (due Apr 11)

Atiyah-Macdonald page 67: 1, 2, 17, 28, 30, 31.

Also prove that the integral closure of R = k[x,y]/(y^2-x^3) is isomorphic to k[t], as follows. Consider the map R -> k[t] given by x|->t^2, y|->t^3. Show that it is injective, so we can consider R as a subring of k[t]. Moreover, they have the same field of fractions. Now on one hand, k[t] is integrally closed, so the closure of R must be included in it. On the other hand, t is integral over R, so k[t] is contained in the integral closure of R.

HOMEWORK 8 (due Apr 12)

Atiyah-Macdonald page 84: 2, 4, 5, 7, 14, 26, 27

Milne page 24: 1.1--1.4

HOMEWORK 9 (due Apr 19)

Milne page 31: 2.1--2.6

HOMEWORK 10 (due May 3)

1. Standard Facts about Finite Fields

Let p be a prime. Observe that if f(x) is an irreducible polynomial of degree d over F_p (the field with p elements), then (F_p)[x]/(f(x)) is a field with p^d elements, which we call F_(p^d).

a. Show that x^(p^d) - x splits into a product of distinct linear factors over F_(p^d) by showing that every element of F_(p^d) is a root of this polynomial.

b. Show that the splitting field of x^(p^d) - x is F_(p^d). Conclude that the field with p^d elements is unique up to isomorphism (i.e., calling it "the field" is justified).

c. Let sigma be the Frobenius automorphism a->a^p. Prove that the Galois group of F_(p^d)/F_p is cyclic of degree d and is generated by sigma. Conclude that the subfields of F_(p^d) are the fields F_(p^k) where k divides d.

d. Prove that x^(p^d) - x factors over F_p as the product of all the monic irreducible polynomials over F_p whose degree divides d. Use this to find the number of irreducible cubic polynomials over F_7.

[Note: as all of these are "standard facts" you can likely look all of them up. Do this only after you have tried to prove them from scratch.]

2. Let k be a field, f be a polynomial of degree n in k[x], and K be the splitting field of f over k.

a. Show that [K:k] divides n!.

b. Show that in order to have [K:k] = n!, it is necessary but not sufficient for f to be irreducible.

3. Find the Galois group of x^7 - 2 explicitly as a permutation group on the roots.

4. a. Show that the splitting field K of x^8-2 is Q(2^(1/8), zeta_8) where zeta_8 is a primitive eighth root of unity.

b. Despite the facts that [Q(2^(1/8)) : Q] = 8 and [Q(zeta_8) : Q] = 4, prove that [K:Q] is actually 16, not 32. (Optional: also explain how you will avoid making similar mistakes in the future, if you have made them in the past.)

c. Find generators for Gal(K/Q) and write explicitly their permutation action on the roots of x^8-2.

5. Let E=k(alpha) where alpha is algebraic over k.

a. If [E:k] is odd, prove that k(alpha^2) = k(alpha).

b. Show more generally that k(alpha^2) = k(alpha) if and only if the minimal polynomial for alpha has an odd-degree term. (In other words, if it has a term b*x^c where b is nonzero and c is odd.)

c. Is it true in general that if m and n are relatively prime and alpha is algebraic of degree m over k, then k(alpha) = k(alpha^n) ?

6. Find the splitting fields and Galois groups of the following polynomials (if they exist):

a. x^3 - 3 over Q.

b. x^3 - x + 1 over Q.

c. x^3 - 3 over Q(sqrt 3).

d. x^3 - 3 over Q(sqrt(-3)).

e. x^4 - 2 over Q.

f. x^4 - 7 over Q.

HOMEWORK 11 (due May 10)

1. a. Find the splitting field K of x^4 - 4x^2 - 1 over Q.

b. Show that Gal(K/Q) is (isomorphic to) the dihedral group of order 8.

c. Find the 8 nontrivial subfields of K and say which 4 of them are Galois over Q.

2. Prove or disprove: Every field of degree 4 over Q has a subfield of degree 2 over Q.

3. Let K/F be an algebraic extension. We say that an element alpha in K is "abelian" if F[alpha] is a Galois extension of F and the Galois group Gal(F[alpha]/F) is abelian. Prove that the set of abelian elements of K is a field.

4. a. Let F < K < L be a tower of field extensions with [L:F] finite, and let alpha be an element of L. If p(x) is the minimal polynomial of alpha over F, prove that K tensor_F F(alpha) is isomorphic to K[x]/p(x) as a K-algebra.

b. Let K1 and K2 be two algebraic extensions of a field K contained in a field L of characteristic zero. Prove that the K-algebra K1 tensor_K K2 has no nonzero nilpotent elements.