Field extension of finite type

A field extension $𝐿 : 𝐾$ is finitely generated iff $𝐿$ is the adjunction of finitely many elements to $𝐾$ , #m/def/field i.e.

𝐿 = 𝐾 (𝛼 1, \dots, 𝛼 𝑛)

for some ${𝛼 𝑖} 𝑛 𝑖 = 1$ . Equivalently, $𝐿 : 𝐾$ is the composite of finitely many intermediate simple extensions.

Properties

Suppose $𝐿 : 𝐾$ is of finite generated, so $𝐿 = 𝐾 (𝛼 1, \dots, 𝛼 𝑛)$

$𝐿 : 𝐾$ is a finite field extension iff it is an algebraic extension iff ${𝛼 𝑖} 𝑛 𝑖 = 1$ are algebraic.

Proof of 1

If $𝐿 : 𝐾$ is finite it then it is automatically algebraic and the generators are also.

Suppose then that $𝐿 : 𝐾$ is algebraic, and thus each of the $𝛼 𝑖$ are algebraic, say of degree $𝑑 𝑖$ . Since intermediate extensions $𝐾 (𝛼 1, \dots, 𝛼 𝑖) : 𝐾 (𝛼 1, \dots, 𝛼 𝑖 - 1)$ are all finite of degree $\leq 𝑑 𝑖$ , it follows so is $[𝐿 : 𝐾] \leq \prod 𝑛 𝑖 = 1 𝑑 𝑖$ , and thus $𝐿 : 𝐾$ is finite.

Other results

Hilbert's Nullstellensatz

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