User:J D Bowen/Math735 HW11

Section 13.4, Problems 5, 6

Section 13.5, Problems 5, 6, 10

5) Let $$F\subset K \ $$ be a finite field extension.  We aim to show $$K \ $$ is a splitting field over $$F \ $$ if and only if every irreducible polynomial in $$F[x] \ $$ that has a root in $$K \ $$ splits completely in $$K[x] \ $$.

Hint: Use theorems 8 and 27 from 13.4

6) Let $$K_1, K_2 \ $$ be finite extensions of $$F \ $$ which are splitting fields over $$F \ $$. We aim to show

(a) Their composite $$K_1K_2 \ $$ is a splitting field over $$F \ $$, and (b) Their intersection $$K_1\cap K_2 \ $$ is a splitting field over $$F \ $$. (Use previous problem)

5)