Math 620 Midterm Exam #2- April 2, 2013 SOLUTIONS
1. Short Answer- no work need be shown. (30 points)
a. Give a polynomial in Q[x] with Galois group isomorphic to the Klein 4-group.
(x2 + 2)(x2 + 5)
b. Give an example of an inseparable eld extension.
Cons
Math 620 Midterm Exam #2- April 2, 2013
1. Short Answer- no work need be shown. (30 points)
a. Give a polynomial in Q[x] with Galois group isomorphic to the Klein 4-group.
b. Give an example of an inseparable eld extension.
c. Give two matrices which have
Math 620 Midterm Exam #1- February 19, 2013
1. Short Answer- no work need be shown. (20 points)
a. Dene isomorphism and equivalence of short exact sequences.
b. Let R be commutative with identity and M, N R-mod. How does one construct an
R-module homomorp
Math 620 Midterm Exam #1- February 19, 2013
1. Short Answer- no work need be shown. (20 points)
f
g
f g
a. Let 0 A B C 0 and 0 A B C 0 be two short exact sequences.
An isomorphism between them is a collection of isomorphisms : A A, : B B
and : C C such t
Math 620 - Midterm Exam #2- April 7, 2009
Instructions: Choose four of the six problems.
1. Give an example of:
a. An inseparable eld extension.
b. A polynomial in Q[x] with Galois group cyclic of order 4.
c. A ring that is not Noetherian.
d. A local ring
1a. Let F = F2 (t). Then the splitting eld of the polynomial x2 t is an inseparable extension
of F .
b. The Galois group of x4 4x2 + 2, which is the minimal polynomial of
order 4 (see homework).
2+
2, is cyclic of
c. Q[x1 , x2 , . . . , xn , . . .] is not
Math 620 Spring 2009- Midterm Exam #1
Instructions: Choose ve of the seven problems.
1. Consider the group G = GL4 (F3 ) where F3 denotes the eld with 3 elements. How many conjugacy classes of elements of order two does G have? Give a representative from
Math 620 Midterm Exam #2- March 30, 2012
1. (20 points) Suppose f (x) K [x] is irreducible in K [x] and L is a eld extension of K
of nite degree relatively prime to the degree of f (x). Prove that f (x) remains irreducible
in L[x].
2. (20 points) Suppose
Math 620 Spring 2012 Midterm 2 Solutions
1. Suppose deg(f (x) = n and let be a root of f (x) in some extension eld. Since f (x)
is irreducible over K we have [K () : K ] = n. Let [L : K ] = m so gcd(n, m) = 1. We
have [L() : K ] = [L() : L] m = [L() : K (
Math 620 Midterm Exam #1- February 27, 2012- SOLUTIONS
1. Short Answer- no work necessary (30 points)
a. Let M be an R module. State the universal property satised by the symmetric algebra S (M ) in terms of an adjoint pair of functors.
Let A be a commuta