**189-570A:** Higher Algebra I

## Assignment 5

## Due: Monday, November 13.

**1**. Let K=Q(t), where t is a root of the equation

t^{3} + t^{2} + t + 2 = 0.

Write (t^{2}+t+1)(t^{2}+t) and
(t-1)^{-1} in the form a+bt + c t^{2}.

**2.**
Let t be the (unique) positive real number such that t^{4}=5.

(a) Show that Q(it^{2}) is normal over Q.

(b) Show that Q(t+it) is normal over Q(it^{2}).

(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.
Show that:

(a) k(u,v) has degree p^{2}
over k(u^{p}, v^{p}).

(b) There are infnitely many distinct extensions containing k(u^{p},v^{p}) and
contained in k(u,v).

**5.**
What are the Galois groups of the following polynomials?

(a) x^{3}-x-1 over Q.

(b) x^{3}-x-1 over Q(s) where s^{2}=-23.

(c) x^{4}-5 over Q. Over Q[t]/(t^{2}-5)?
Over Q[t]/(t^{2}+5)? Over
Q[t]/(t^{2}+1).

(d) x^{n}-t, over C(t).

(e) x^{4}-t, over R(t).

**6.**
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]/(x^{2}- (a-bs))

(iii) Either a^{2}-cb^{2} or
c(a^{2}-cb^{2}) is a perfect square in k.

Show that L is cyclic over k if and only if
c(a^{2}-cb^{2}) is a square in k.

**7.**
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 S_{n} 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
identity.

(c) If n > 3, show that a^{n} does not belong to Q.

**8.**
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.

Show that

(a)
G is isomorphic to the quotient of
GL_{2}(F_{p}) by its center, and hence has
order p^{3}-p.

(b) The fixed field of G is equal to k(Y), where

Y = (t^{p2}-t)^{p+1}/
(t^{p}-t)^{p2+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*.