Math 544 Qualier Review Problems 11
Unless ortherwise stated, R is a commutative ring with identity and all Rmodules are unitary.
1. Let M be a Noetherian R-module and u : M M a surjective homomorphism of R-modules. Show that u is an isomorphism.
Proof. N
Math 544 Qualier Review Problems 10
1. Find the Galois group of the following quintics over Q
(a) X 5 4X + 2
(b) X 5 6X + 3
Proof.
(a) X 5 4X + 2 is irreducible by Eisensteins criterion with p = 2. From
calculus, X 5 4X + 2 has three real roots and one pa
Math 544 Qualier Review Problems 9
1. Determine the Galois groups of the following polynomials over Q
(a) X 3 3X + 1
(b) X 3 + 3X 2 X 1
(c) X 4 4X 2 + 5
Proof.
(a) Note that X 3 3X + 1 is irreducible over Q by the rational root
test. The cubic is already
Math 544 Qualier Review Problems 8
1. Let p be a prime and let n be an integer greater than 1. Suppose K/F is a
Galois extension of elds, with Gal(K/F ) Cpn . Let L be an intermediate
=
eld such that [K : L] = p.
(a) Suppose E = K is an intermediate eld i
Math 544 Qualier Review Problems 7
1. Let n > 2 and let be a primitive nth root of unity. Show that Q( )
contains a subeld E such that E R and [Q( ) : E ] = 2.
Proof. Consider + 1 = 2Re( ). Then E = Q( + 1 ) R is intermediate in the extension Q( )/Q. What
Math 544 Qualier Review Problems 6
1. Prove that K is a splitting eld over F of a nite set of polynomials
cfw_f1 , . . . , fn in F [X ] if and only if K is a splitting eld over F of the
single polynomial f = f1 f2 fn F [X ].
Proof. Suppose rst that K is
Math 544 Qualier Review Problems 5
1. (a) Find the minimum polynomial of =
3+
5+
(b) Find the minimum polynomial of =
(c) Find the minimum polynomial of =
(d) Find the minimum polynomial of =
3
7 over Q.
3 over Q.
3 + 2 2 over Q.
2 + 2 over Q.
Proof.
7 an
Math 544 Qualier Review Problems 4
1. Let F be a eld and F its multiplicative subgroup. Show that the Abelian
groups (F, +) and (F , ) are not isomorphic.
Proof. We consider two cases. Suppose rst that char F = 2. Then F
has at least 2 elements, and all n
Math 544 Qualier Review Problems 3
Unless ortherwise stated, R is a commutative ring with identity and all Rmodules are unitary.
1. Suppose the diagram below commutes, where the horizontal maps are
R-module homomorphisms, and the vertical maps are R-modul
Math 544 Qualier Review Problems 2
Unless ortherwise stated, R is a commutative ring with identity and all Rmodules are unitary.
1. Suppose A and B are Abelian groups and m and n are integers such
that mA = 0 = nB . Show that every element of HomZ (A, B )
Math 544 Qualier Review Problems 1
Unless ortherwise stated, R is a commutative ring with identity and all Rmodules are unitary.
1. Let F be a eld and let R = Matn (F ).
(a) Show that the set of n-tuples F n can be viewed as an R-module under
the natural