Field theory MOC

Adjoining a root to a field

Let be a field and be a nonzero ^irreducible. Then

is a simple extension field of , with primitive element . Moreover, if is a field extension so that has a root in , then we have a tower of field extensions .¹ field

Proof

Since is a Euclidean domain and thus in particular a PID. By Maximal ideal iff prime ideal in a PID, it follows is maximal and thus as defined is indeed a field (Condition for a quotient commutative ring to be a field). Let denote the projection. Then

Since all we adjoined was , this is indeed simple.

Now suppose is an extension with , , so the Evaluation map vanishes at , whence and thus by the universal property of quotients there is a unique homomorphism

which gives the desired tower of extensions.

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Footnotes

1. 2009. Algebra: Chapter 0, §V.5.2, pp. 283–284 ↩