ALGEBRA 612. FINAL EXAM.
Name:
The test contains three sections (Galois Theory, Commutative Algebra,
and Representations of Finite Groups). Each section contains three problems. You have to choose two problems from each section. Only these
problems will b
ALGEBRA 612, SPRING 2010. HOMEWORK 3
1. Let p1 , . . . , pr Z be distinct primes and let K = Q( p1 , . . . , pr ). (a) Compute
the Galois group Gal(K/Q). (b) Describe explicitly all intermediate subelds L
such that either [L : Q] = 2 or [K : L] = 2. (c) D
ALGEBRA 612, SPRING 2010. HOMEWORK 2
In this set we x a nite eld extension K F . Let K be an algebraic
closure of K.
1. Let F and let f (x) be its minimal polynomial. Suppose that is
not separable over K. (a) Show that char K = p and f (x) = g(xp ) for so
ALGEBRA 611, SPRING 2010. HOMEWORK 1
(1)
In this set we x a eld extension K F .
1. Let R be an innite domain and let f R[x]. Prove that f (r) = 0 for
innitely many r R. What if R is not necessarily a domain?
2. (a) Show F is algebraic over K if and only i
ALGEBRA 612, SPRING 2010. HOMEWORK 5
In this worksheet, k denotes a eld. Do not assume that k is algebraically
closed unless otherwise stated.
1. Let S be a nite set and let I be a non-empty collection of its subsets
(called the independent sets). A pair
ALGEBRA 612, SPRING 2010. HOMEWORK 4
1. Show that Sn is solvable if and only if n 4.
2. (a) Let f (x) K[x] be an irreducible separable polynomial with roots
= 1 , 2 , . . . , n K.
Suppose that there exist rational functions 1 (x), . . . , n (x) K(x) such
ALGEBRA 612, SPRING 2010. HOMEWORK 8
In this worksheet the base eld is always C unless otherwise stated.
1. Describe explicitly (i.e. not just dimensions but how each element of the
group acts) all irreducible representations of (a) (Z/2Z)r ; (b) D2n .
2.
ALGEBRA 612, SPRING 2010. HOMEWORK 6
In this worksheet, k denotes a eld and R denotes a commutative ring.
Do not assume that k is algebraically closed unless otherwise stated.
1. Let S R be a multiplicative system. Consider the covariant functor Rings Set
ALGEBRA 612. TAKE-HOME MIDTERM.
The take-home midterm is due before class on Wednesday Apr 21. You
can use lecture notes and a textbook (Knapp, or AtiyahMacdonald, or any
other textbook of your choosing). You should work independently: no outside help is
ALGEBRA 612, SPRING 2010. HOMEWORK 7
In this worksheet R is a ring and k = k is an algebraically closed eld.
1. Let f, g k[x1 , . . . , xn ] be polynomials such that f is irreducible and
V (f ) V (g). Show that f divides g.
2. Let : A2 A2 be a morphism gi