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189-570A: Higher Algebra I

Assignment 5

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.

2. 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. Show that:

(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).

5. 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 Q[t]/(t2+1).

(d) xn-t, over C(t).

(e) x4-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]/(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.

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 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 identity.

(c) If n > 3, show that an 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 GL2(Fp) by its center, and hence has order p3-p.

(b) The fixed field of G is equal to k(Y), where
Y = (tp2-t)p+1/ (tp-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.