1. T/F
a. A splitting field over K for a polynomial of degree n will have degree n! over K. T F
b. If [E:K] = n, then Gal(E/K) has n elements. T F
c. If [E:K]s = [E:K], then E is separable over K. T F
d. Every extension of the field of rationals is separable. T F
e. Every extension field of degree 2 over K with ch K ¹ 2 is normal. T F
f. If [E:K] = n finite with E separable over K, then E = K(a ) for some a in E. T F
g. If E = K(a ) where ch K = p and a Ï K. ap Î K, then Gal(E/K) has order p. T F
h. Every UFD is a PID. T F
i. All algebraic extensions E of a given field K are finite extensions. T F
j. Q(Ö 2) @ QÖ 3) just as fields. T F
k. One can adjoin a primitive 5th root of unity to Z5. T F
l. If K Í F Í E, then the fixed field of Gal(E/F) is F. T F
m. If K Í F Í E, then the subgroup of Gal (E/K) which fixes F is Gal(E/F). T F
n. If K Í F Í E, and E is Galois over K, then F is Galois over K. T F
o. If K Í F Í E, and E is Galois over K, the E is Galois over F. T F
p. Q(x ) is a cyclic extension of Q where x is a primitive eigth root of unity. T F
q. All intermediate rings between Z and Z[X,Y] are finitely generated over Z. T F
r. For every permutation of the roots of an irreducible polynomial p(x) in K[x], there is an automorphism of K({roots of p(x)}) which matches the permutation applied to the roots. T F
s. If E Ê K, every k homomorphism of E is an isomorphism. T F
t. If E is algebraic over K, then E is isomorphic to some subfield of K. T F
u. EF exists where E = Reals, and F = Zp where p is a prime. T F
v. If E is a field, then E can have subrings which are not fields. T F
w. Every subring with one of a field is an integral domain. T F
x. There exist finite integral domains which are not fields. T F
y. Every nonzero prime ideal of K[X] where K is a field is a maximal ideal. T F
z. Every subfield of E = K(X) which properly contains K is isomorphic to E. T F
aa. All fractional ideals in a Dedekind domain are 2-generated. T F
bb. If A and B are fractional ideals in an integral domain D, then A: B is also a fractional idea. T F
cc. D[X] is a UFD iff D is a UFD. T F
dd. R[X] has the following universal property. If there exists a ring homomorphism q from R to another ring S and s is any element of S, there exists a unique extension of q from R[X] to S which takes X to s. T F
ee. Unique factorization holds for subrings of the complexes which contain Z. T F
ff. No quintic polynomial is solvable by radicals. T F
2. Are there any false statements above will be true if the requirement
of E Galois over K iis added? If so, which ones?
3. Compute the matrix which accomplishes the linear transformation of
multiplying by Ö 6 in the Q vector space
E = Q[Ö 2, Ö
3 ] and compute the norm and trace of Ö
6 in E over Q.
4. Let K Í F Í E.
Consider the open sentence : E is X over K iff F is X over K and E is X over F.
For which X is the statement true and which false?
i) X = finite
ii) X = algebraic
iii) X = purely inseparable
iv) X = separable
v) X = normal
vi) X = Galois
vii) X = solvable by radicals
viii) X = finitely generated
5. Find an extension of Q whose Galois group is Z2 x Z2.
6. Let K have characteristic p, and E is an algebraic extension of K.
Prove that if a in E is both separable and purely
inseparable over K, that a is in K.