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).
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
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 subel
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 ove
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
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 collectio
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
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 ir
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 multipli
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 choosi
Take-Home Quiz PPT 9 *Nomenclature*Dr. Commodari *The Ursuline School* Due:03/01/18
Nam:
Name: _
Matching
Match each item with the correct statement below.
a. monatomic ion
f.
b. acid
g.
c. base
h.
d.