Adjoining a root to a field

Let $𝐾$ be a field and $𝑓 (𝑥) \in 𝐾 (𝑥)$ be a nonzero irreducible polynomial. Then

𝐾 (𝛼) : = 𝐾 [𝑥] ⟨ 𝑓 (𝑥) ⟩

is a simple extension field of $𝐾$ , with primitive element $𝛼 = 𝜋 (𝑥)$ . Moreover, if $𝐿 : 𝐾$ is a field extension so that $𝑓 (𝑥)$ has a root in $𝐿$ , then we have a tower of field extensions $𝐿 : 𝐾 [𝛼] : 𝐾$ .¹ #m/thm/field

Proof

Since $𝐾 [𝑥]$ is a Euclidean domain and thus in particular a PID. By Maximal ideal iff prime ideal in a PID, it follows $⟨ 𝑓 (𝑥) ⟩$ is maximal and thus $𝐾 (𝛼)$ as defined is indeed a field ( $𝑅 / 𝐼$ for commutative $𝑅$ is a field iff $𝐼$ is maximal). Let $𝜋 : 𝐾 [𝑥] ↠ 𝐾 (𝛼)$ denote the projection. Then

𝑓 (𝜋 (𝑥)) = 𝜋 (𝑓 (𝑥)) = 0 .

Since all we adjoined was $𝛼 = 𝜋 (𝑥)$ , this is indeed simple.

Now suppose $𝐿 : 𝐾$ is an extension with $𝑓 (𝛽) = 0$ , $𝛽 \in 𝐿$ , so the Evaluation map $𝜖 (𝛽) : 𝐾 [𝑡] \to 𝐿$ vanishes at $𝑓 (𝑥)$ , whence $⟨ 𝑓 (𝑥) ⟩ \leq k e r 𝜖 (𝛽)$ and thus by the universal property of quotients there is a unique homomorphism

𝑗 : 𝐾 (𝛼) \to 𝐿

which gives the desired tower of extensions.

#state/tidy | #lang/en | #SemBr

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

Adjoining a root to a field

Footnotes