
76 Simple Extensions
It is clearly onto, and so is an isomorphism. Further, φ |
K
is the identity, so that φ
defines an isomorphism of extensions. Finally, φ (t) = α.
The classification for simple algebraic extensions is just as straightforward, but
more interesting:
Theorem 5.12. Let K(α) : K be a simple algebraic extension, and let the minimal
polynomial of α over K be m. Then K(α) : K is isomorphic to K[t]/hmi : K. The
isomorphism K[t]/hmi → K(α) can be chosen to map t to α (and to be the identity
on K).
Proof. The isomorphism is defined by [p(t)] 7→ p(α), where [p(t)] is the equivalence
class of p(t) (mod m). This map is well-defined because p(α) = 0