This file includes some HTML versions of previous finals exams I have given in Math 421 (Abstract Algebra II). Note that it uses HTML tags for subscripts and superscripts, so it is difficult to decipher with an earlier viewer.
Each question is worth 25 points. Each definition is worth 5 points.
1.
(a) Define: normal subgroup.
(b) In the dihedral group D8 of order 16,
let H={e,a2,a4,a6}.
Is H a normal subgroup of D8?
(c) In D8, let K={e,a4,b,a4b}.
Is K a normal subgroup of D8?
2.
Let G = Z17x,
the multiplicative group of units of the field Z17,
and let N=<4>={1,4,13,16}
(a) Show that N is a subgroup of G.
(b) List the cosets of N in G.
(c) Find the order of each element of G/N.
3.
(a) State the division algorithm for polynomials over a field F.
(b) State Eisenstein's irreducibility criterion.
(c) State Kronecker's theorem on the existence of roots.
4.
(a) Define: integral domain; prime ideal; maximal ideal.
(b) Give an example of a commutative ring that is not an integral domain.
(c) Give an example of a prime ideal that is not a maximal ideal.
5.
(a) Define: minimal polynomial of an element in an extension field.
(b) Show that Q(sqrt(5)+i)=Q(sqrt(5),i).
(c) Find the minimal polynomial of sqrt(5)+i over Q.
6.
(a) Define: finite extension field; algebraic element; algebraic extension field.
(b) Prove that any finite extension field is an algebraic extension.
7.
Let R and S be commutative rings (with multiplicative identity elements).
(a) Define the addition and multiplication of elements in the
direct sum R(+)S of R and S.
(b) Show that if I is an ideal of R, and J is an ideal of S, then
I(+)J = { (x,y) | x in I and y in J }
is an ideal of R(+)S.
(c) Prove that (R(+)S) / (I(+)J) is isomorphic to R/I (+) S/J.
8.
Let G be a group, let N be a normal subgroup of G, and
let H be any subgroup of G.
(a) Assuming that the intersection of H and N is a subgroup,
show that it is normal in H.
(b) Prove that
HN = { x in G | x = yz for some y in H, z in N } is a subgroup of G.
(c)
Prove that
(HN) / N is isomorphic to H / (H intersect N).
Bonus question: Prove that if E is an algebraic extension of K and F is an algebraic extension of E, then F is an algebraic extension of K.
Each question is worth 25 points. Note the choice in questions 7 and 8.
(1) State the definitions of normal subgroup and left coset. Show that if a subgroup N of a group G is normal, then multiplication of left cosets is well-defined.
(2) State the Fundamental Homomorphism Theorem for groups, and use it to prove that any cyclic group is isomorphic to either Z or Zn, for some n.
(3)
Assume that the dihedral group D4 is given as
D4 =
{ 1e, a, a2, a3, b, ab, a2b, a3b },
where a has order 4, b has order 2, and ba=a-1b,
and let N = {e, a }.
Show by a direct computation that N
is a normal subgroup of D4.
Is the factor group D4 / N a cyclic group?
(4) Construct a field with 8 elements.
(5)
Let R be a commutative ring with identity 1.
For any element a in R, show that
Ann (a) = { r in R | ra } =0
is an ideal of R.
Find Ann (a) for the element a = (0,2)
in the direct sum of the rings
Z12
and Z8.
(6) State the definitions of finite extension field, algebraic element, and algebraic extension field. Prove that any finite extension field is an algebraic extension.
(7) Choose either A or B.
A. Let K be a subfield of E, and let E be a subfield of F. Prove that if E is an algebraic extension of K and F is an algebraic extension of E, then F is an algebraic extension of K.
_ _
B. Show that Q(\/2+i) = Q(\/2,i).
_
Find the minimal polynomial of \/2+i over Q.
(8) Choose either A or B.
A. Prove that if F is a field, then the ring of polynomials F[x] with coefficients in F is a principal ideal domain.
B. Prove that any nonzero prime ideal of a principal ideal domain is maximal.
Results:
13 students took the test; the high score was 187/200, the low score 68/200. The mean was 110, with a standard deviation of 36.
(1)
(a) Define: normal subgroup, left coset, right coset.
(b) Show that a subgroup is normal if and only if
its left and right cosets coincide.
(c) Show that if N is a normal subgroup of G,
then coset multiplication is well-defined.
(2)
Let G be the dihedral group of order 8, given by
G =
{ e, a, a2, a3, b, ab, a2b, a3b },
where a has order 4, b has order 2, and ba=a-1b,
and let H = {e, a }.
(a) Show that H is normal in G.
(b) To what familiar group is G / H isomorphic?
(3)
(a) State the Fundamental Theorem of Group Homomorphisms.
(b) Use the theorem to prove that any cyclic group is isomorphic
to either the group
Z
of integers or the group
Zn
of integers modulo n,
for some n.
(c) Find all group homomorphisms from the group
Z9
of integers modulo 9 into the group
Z12
of integers modulo 12.
(4) CHOOSE either A or B:
A:
(a) Define integral domain, ideal, principal ideal.
(b) Prove that if F is a field,
then F[x] is a principal ideal domain.
B:
(a) Define what it means for F to be
an extension field of K.
If u is an element of F,
give the definition of the minimal polynomial
of u over K,
and the definition of the degree
of u over K.
(b) Let F be an extension field of K,
and let u be an element of F.
Prove that if p(x) is the minimal polynomial
of u over K,
then p(x) | f(x) for any
f(x) in K[x] with f(u) = 0.
(5)
(a) Define: finite extension,
algebraic element, algebraic extension.
(b) Prove that any finite extension is an algebraic.
(c) Prove that if
F is a finite extension of E, and
E is a finite extension of K, then
F is a finite extension of K, and
[F:K] = [F:E] [E:K].
(6) Let u be the complex number defined by
___
3 /
u = \/ 2 + i .
(a) Find the degree of u over Q.
(7) CHOOSE either A or B:
A. State necessary and sufficient conditions under which a real number is constructible. Outline the proof that there exists an angle that cannot be trisected with straightedge and compass.
B.
Let G be a group, and let D be the smallest subgroup
which contains all elements of the form
xyx-1y-1, for some x,y in G.
Prove that D is a normal subgroup,
and that G / D is abelian.
Results:
9 students took the test; the high score was 188/200, the low score 84/200. There were 3 A's, 1 B, and 5 C's.
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