MATH 210B
ASSIGNMENT 3
(Due Fri, February 3)
1. If p is an odd prime, show that the multiplicative group of units in (Z/p n)
is cyclic for all n. Thus Aut(Z/pn) = (Z/pn)* is cyclic.
2. What is the order of 1+4 in the multiplicative group (Z/2n)* for n  3?
Show that –1 is not a power of 1+4 in (Z/2n)* for n  3. Show that (Z/2n)* =
Z/2  Z/2n-2 for n 3.
3. Find a vector space basis for C(z) over C, where C(z) is the field of
rational functions in one variable over the complex numbers. What about
Q(z) over Q?
4. Prove that if K is separable over F and E is separable over K, then E is
separable over F, as claimed in Paragraph 8 of the Galois theory outline.
5. Prove the result in Paragraph 16(ii) in the Galois theory outline.
6. Prove the results in Paragraph 17(i) of the Galois theory outline.
7. Show that if F is a field, p is a prime, and a  F, then the polynomial
Xp – a  F[X] is either irreducible or has a root in F.
8. Suppose the pth roots of unity, i, 0 i  p-1, p prime, are in a field F and
suppose char(F)  p. If F  E = F[] is a cyclic Galois extension of degree
p, let j = j(), 0 j p-1, denote the conjugates of , where <> = Gal(E/F),
and for each i, let i denote the Lagrange resolvant i = 0jp-1 ijj. If 
and  denote the column vectors with entries i and i, respectively, find a
matrix M over F with M = . Deduce that for some i  1, i  F, and that
for such, ip  F and E = F[i]. (Note 0 =  j  F.)
9. Let E/k be a finite Galois extension with group G = Gal(E/k). Define the
trace T:E  E by T(u) =  G (u). Prove that T is k-linear and that
im(T) = k.
10. In the situation of Problem 9, suppose G = <> is a cyclic group and
define  =  - Id:E  E. Show ker() = k and im() = ker(T). Thus, if v  E
one has T(v) = 0 if and only if v = (u) – u for some u  E.
HINTS FOR ASSIGNMENT 3
1. (Z/pn)* projects onto (Z/p)*, which is cyclic. Determine the order of 1+p
in (Z/pn)* by induction.
2. Induction.
3. Partial fractions.
4. Count embeddings. I think you need Problem 5 from Assignment 2.
5. You need to use the fact that each element of E satisfies a separable
equation of degree  |H| over EH and then use Primitive Element to prove
that by adjoining a finite list of elements in E to EH, you can’t get a degree
bigger than |H|.
6. You first need finite, normal, separable, all of which are easy. Then
define an obvious injection Gal(KL/L)  Gal(K/F). Finally figure out why
the image is as claimed.
7. There are two cases, depending on the characteristic of F. In both cases,
the key is showing that if Xp – a has a factor of degree d < p, then the dth
power of some root lies in F.
8. Vandermonde.
9. You need to use linear independence of characters if the characteristic of
k is not 0.
10. Use k = E<>, a vector space inclusion, problem 9, and some dimension
counts.