322 MATH Final Examination
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Question
1
2
3
4
June 2, 2001
9:0012:00
TB 120
Score
/10 points
/15 points
/5 points
/5 points
5
6
7
8
9
/5
/5
/10
/15
/10
points
points
points
points
points
10
Bonus Question
/20 points
/20 points
Total
/100 points
1. Express the symmetric polynomial x3 + y 3 + z 3 + xyz over Z in terms of the
elementary symmetric polynomials.
(10 points)
1
2. Let
A :=
a
c
b
d
Mat22 (C) :
a
c
b
d
0i
11
=
0
1
i
1
a
c
b
d
Mat22 (C).
Is A a vector space over R? Over C?
(15 points)
2
3. Let K be a eld. Give the denition of the characteristic of K .
3
(5 points)
4. Let K be a eld of characteristic p = 0. Prove that : K K , a a p is a
eld homomorphism.
(5 points)
4
5. Find the multiplicative inverse of 2 + 3 5 in the eld F7 ( 5).
5
(5 points)
6. Let E/K be a eld extension and let D be integral an domain such that K
D E . Prove that, if E is algebraic over K , then D is a eld.
(5 points)
6
7. Find the number of monic irreducible polynomials of degree 72 over F 5 .
(10 points)
7
8. Let E/K be a eld extension and let L be an intermediate eld. If L is
(K, E )-stable, prove that L is (K, E )-stable, too.
(15 points)
8
9. Find an algebraic eld extension that is not separable.
9
(10 points)
2i
10. Let := e 12 C and E := Q( ). Find all subgroups of G := AutQ E , all
intermediate elds of E/Q, and a primitive element for each of the intermediate
elds. Describe the Galois correspondence by Hasse diagrams of groups and intermediate elds.
(20 points)
10
Bonus question. Tell about the contribution to algebra of one of the following
mathematicians: Gauss, Cauchy, Abel, Liouville, Klein, Kronecker, Dedekind,
Heinrich Weber, Artin.
11