HW due May 4

1. Let F be Galois over K, and f in F be algebraic over K. Let {f_i} be the orbit of f under the Galois group.
a. Show that the set {f_i} is finite, and
b. the polynomial \Prod (x-f_i) is in K[x], irreducible, and has f as a root.

2. Let S_n act on the field Q(a₁,...,a_n) of rational functions in n variables. Let e_i^S be the ith elementary symmetric polynomial in the {a_j, j in S}. As that Wikipedia page (v. 4/28/07) indicates, the invariant subfield is generated by the {e_i^1,2,...,n =: e_i}.

a. What is the minimal polynomial of a₁, in Q(e₁,...,e_n)?
b. Let n=4. Describe all subgroups of S₄ up to conjugacy.
For each conjugacy class of subgroup of S₄, write down generators for the fixed field.
For each such field F, write down the minimal polynomial in F[x] for each a_i.

3. Let F be a field of characteristic p.
a. If F is finite, show that the Frobenius is onto.
b. If F is algebraically closed, show that the Frobenius is onto.
c. Give an example where it's not onto.

Fri Apr 20

Finite fields are Galois over F_p. The Galois group is generated by the Frobenius.

Nailed down the Galois correspondence. Lemma: If F > E > H > K, E > H is finite, and H''=H, then E''=E. Galois correspondence when applied to H=K.

An extension is Galois iff the Galois group is big enough.

Normal subgroups <-> subfields E stable under the Galois group, which in turn implies that F/E, E/K are Galois extensions, and there is a short exact sequence of Galois groups.

HW due Fri Apr 27

1. Let F > K be an algebraic field extension, f in F, p(x),q(x) two nonzero polynomials in K[x] of which f is a root. Assume that p(x) has lowest degree. Show that p is irreducible and p|q.


2. Define the derivative q' of a polynomial q without using limits, so that you can apply it to polynomials with arbitrary coefficients.

a. If p,q are two polynomials such that p² | q, show that p divides gcd(q,q').

b. If q is an irreducible polynomial in K[x] where K is a field of characteristic zero, and F is a field extension of K, show that q cannot have a repeated factor when factored in F[x].


3. Let F = F_p(a), the field of rational functions. Consider the map F[x] -> F_p(b) taking x |-> b, a |-> b^p.

a. Prove that x^p-a is irreducible in F[x].
Hint. For r(b) in F_p[b], show that r(b^p) = r(b)^p, a nice extension of the Freshman's Dream. Then reconsider the map above.

b. Let F_p(b) be the extension suggested above, where b^p = a. Show that b is the only root of the irreducible polynomial x^p-a.


4. Let p(x) = the polynomial x^pn - x in F_p[x].

a. Show that p(x) has no repeated roots.

b. Show that p(x) is the product of all monic irreducible polynomials in F_p[x] of degree at most dividing n.

c. Write out this factorization for p=2, n=4.

HW due Friday April 20

1. Show that Q[cube root of two, omega] is a simple extension, where omega = exp(2 pi i/3).

2. Let F = K[u], a finite simple extension, where u satisfies an irreducible polynomial p(x) in K[x].

a. Let u' be another root in F. Show there exists an automorphism of F taking u to u'.

b. Show |Gal(K/F)| is at most deg(f) deg(p).

c. Determine Gal(Q[cube root of two, omega] / Q), meaning, for each group element write down what it does to those two field elements.

3. Let F > K denote a field extension, with R a ring in between the two.

a. Give an example where R is not a field.

b. If F is algebraic over K, show R is indeed a field.

4. Let R be a domain, K the fraction field of R (so, not matching the notation in #3), F an extension of K, and f in F algebraic over K.
Show there exists a nonzero r in R such that rf is integral over R.

5. What are all the rings between Z and Q?
Which ones are finitely generated as Z-algebras?
Which ones are finitely generated as Z-modules?

Fri Apr 6

This was a one-shot lecture in case prospective students showed up.

Composites of left adjoints are left adjoints.
In particular, the symmetric algebra of a module is a quotient of the tensor algebra.
The universal enveloping algebra of an anticommuting algebra.
The Jacobi identity.
Lie algebras.
Statement of the Poincare-Birkhoff-Witt theorem.

HW #1 due Friday Apr 13

1. In class we lamented that for G acting on a finite set X, we could hardly figure out anything about the action knowing only X^G.

Let Z^X be the free abelian group generated by X; the action of G on X extends in a unique way to an action on Z^x by group automorphisms.

What can you determine about the action of G on X, knowing (Z^X)^G?

2. Define an R-supermodule as...
oh, this sounds so exciting! ...
... a pair (M₀,M₁) of modules. And a morphism thereof is just a pair of morphisms of R-modules. Pretty pedestrian really.
There are several forgetful functors R-SuperMod -> R-Mod, but the one we want is (M₀,M₁) |-> M₀ + M₁ (direct sum).

Slightly more interesting: if R is commutative, so that the tensor product of two R-modules is naturally an R-module again, define
(M₀,M₁) @_R (N₀,N₁) = (M₀ @_R N₀ + M₁ @_R N₁, M₀ @_R N₁ + M₁ @_R N₀).

a. Make (detailed) sense of the following statement, and prove it (not much detail):
"The functors Forget o @ and @ o Forget are naturally isomorphic."

Define an R-superalgebra as an R-supermodule M plus an associative "multiplication" M @_R M -> M.

b. Given an R-module N, define a superalgebra structure on the supermodule (R,N).

Call a superalgebra (M₀,M₁) supercommutative if M₀ commutes with everything, and M₁ anticommutes with itself, i.e. ab = -ba for a,b in M₁. (Cohomology rings from topology are examples, where M₀ is the sum of the even-degree parts, M₁ the odd.)

c. Given a supermodule M, define the "free supercommutative superalgebra" on M. It should be a left adjoint to the forgetful functor from supercommutative superalgebras to supermodules.

3. Let R = Q[x] / < x² - n >, where Q is the rationals and n is an integer. Assume R is not a field. Describe R in terms of more familiar rings.

4. Let H be a subgroup of G. Show there is a unique largest normal subgroup of H and give a way to check whether h in H is an element thereof.