2. F is a field. Show that every finite subgroup H of (F-{0},x) is cyclic.
Solution

3. Let F be a field with characteristic <> 2.
(a) If E is a quadratic extension of F (meaning [E:F]=2), then prove that Delta(E)={x in (F,x);
there exists a in E, x=a^2 } is a subgroup of (F,x) containing (F,x)^2.
(b) If E and E' denote two quadratic extensions of F, then Delta(E)=Delta(E') => E and
E' are F-isomorphic.
(c) Prove the existence of infinitely many quadratic extensions of Q which are pairwise non-isomorphic.
(d) Prove that there is only one field with order p^2.
In the notations above, (F,x) denotes the multiplicative group of F.
Solution

4. If F is a perfect field, then every algebraic extension K of F is perfect.
Solution

5. (a) Consider a field F with characteristic p. Prove that if the polynomial P(X)=X^p-X-a is
reducible in F[X], then P splits into distinct factors in F[X].
(b) Show that X^p-X-1 is irreducible in Q[X].
Solution

6. Give the degree over Q of a splitting field S of the polynomial P(X)=X^5-3 of Q[X].
Solution

7. F_2=Z/2Z. Given K=F_2(x) and L=F_2(sqrt(x)), show that the trace map
T^L_K(u) defined by the trace of the K-linear map L -> L: x -> ux is identically 0.
Solution

8. t is transcendental over F and we note K=F(t). Consider a field E such that E <> F and
F C E C K. Prove that t is algebraic over E.
Solution

9. Let K=Q(i,sqrt(2)). Prove that K/Q is a Galois extension, and determine its
Galois group. Determine if K(sqrt(1+sqrt(2))) is a Galois extension of Q.
Solution

10. Let f(X) in Q[X] be a polynomial irreducible over Q, with complex roots
alpha and beta. Show that, if E is a finite Galois extension of Q (Q C E C C),
then Q[alpha] Intersect E and Q[beta] Intersect E are isomorphic. Prove that if E=Q[epsilon] where
epsilon is a root of unity, then Q[alpha] Intersect E = Q[beta] Intersect E.
Solution