189-570A: Higher Algebra I
Due: Monday, November 13.
1. Let K=Q(t), where t is a root of the equation
t3 + t2 + t + 2 = 0.
Write (t2+t+1)(t2+t) and
(t-1)-1 in the form a+bt + c t2.
Let t be the (unique) positive real number such that t4=5.
(a) Show that Q(it2) is normal over Q.
(b) Show that Q(t+it) is normal over Q(it2).
(c) Show that Q(t+it) is not normal over Q.
3. Let K be a field.
Prove that every algebraic extension of K is seperable, if and only if
K has characteristic 0, or K has characteristic p and every element of
K has a pth root in K.
4. Let k be a field of characteristic p and let u, v be indeterminates.
(a) k(u,v) has degree p2
over k(up, vp).
(b) There are infnitely many distinct extensions containing k(up,vp) and
contained in k(u,v).
What are the Galois groups of the following polynomials?
(a) x3-x-1 over Q.
(b) x3-x-1 over Q(s) where s2=-23.
(c) x4-5 over Q. Over Q[t]/(t2-5)?
Over Q[t]/(t2+5)? Over
(d) xn-t, over C(t).
(e) x4-t, over R(t).
Let k be a field of characteristic different from 2, and let
c be an element of k which is not a square in k.
Let K/k be the quadratic extension
of k obtained by adjoining to k a square root s of c.
Let L/K be the quadratic extension obtained by adjoining to K a square
root of a+bs.
Show that the following are equivalent:
(i) L is Galois over k.
(ii) L is isomorphic to K[x]/(x2- (a-bs))
(iii) Either a2-cb2 or
c(a2-cb2) is a perfect square in k.
Show that L is cyclic over k if and only if
c(a2-cb2) is a square in k.
Let f be a polynomial over Q of degree n, and let K/Q be its splitting
field. Suppose that Gal(K/Q) is isomorphic to the symmetric
group Sn with n > 2.
(a) Show that f is irreducible over Q.
(b) If a is a root of f, show that the only automorphism of Q(a) is the
(c) If n > 3, show that an does not belong to Q.
Let k be the finite field with p elements (p a prime).
Let K=k(t) be the field of rational functions
in the variable t. Let G be the
group of automorphisms of K obtained by the mappings
t |--> (at + b) / (ct+d), with a,b,c,d in k, ad-bc non-zero.
G is isomorphic to the quotient of
GL2(Fp) by its center, and hence has
(b) The fixed field of G is equal to k(Y), where
Y = (tp2-t)p+1/
(c) k(t) is a Galois extension of k(Y) with Galois group G.
It follows from this discussion that the group
G arises as the Galois group of a finite extension of the
field of rational functions k(x).
9. Let K be a field and let f be a polynomial in K[x].
Show that the splitting field of f hose existence was proved in class
is unique up to isomorphism.