Primitive Element Theorem

Theorem

Let $E / F$ be a separable field extension of finite degree.

Then $E / F$ is simple: there exists $\alpha\in E$ such that $E = \map F \alpha$.

Proof

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If $F$ is a finite field (equivalently $E$ is a finite field), this follows from Finite Extension of $\F_p$ is Generated By a Single Element, since the generator of $E / \F_p$ also generates $E / F$.

Next, assume $F$ is infinite.

Choose an algebraic closure $\overline F$ of $F$.

Let $\sigma_1, \dots, \sigma_n$ be all distinct embeddings of $E$ into $\overline F$ fixing $F$.

By definition 2 of separable degree,

$n = \index E F_s$

Since $E / F$ is separable,

$n = \index E F_s = \index E F$

Note that if $i \ne j$, then $\sigma _ i \ne \sigma _ j$, so

$V_{ij} = \map \ker {\sigma _ i - \sigma _ j}$

is not equal to $E$.

By Vector Space over an Infinite Field is not equal to the Union of Proper Subspaces,

$\ds \bigcup_{1\le i < j\le n} V_{ij}$

is a proper subfield of $E$,

so we can find $\alpha \in E$ with $\alpha \notin V_{ij}$ for all $i \ne j$.

It follows that $\map {\sigma_1} \alpha, \dots, \map {\sigma_n} \alpha$ are distinct,

so there are at least $n$ embeddings of $F \sqbrk \alpha$ into $\overline{F}$ fixing $F$,

so

$\index {F \sqbrk \alpha} F_s \ge n$

but since $F \sqbrk \alpha / F$ is a subextension of $E / F$, by Subextensions of Separable Field Extension are Separable, $F \sqbrk \alpha / F$ is separable,

so

$\index {F \sqbrk \alpha} F = \index {F \sqbrk \alpha} F_s \ge n$

On the other hand, by Tower Law,

$\index {F \sqbrk \alpha} F \le \index E F = n$

Hence

$\index {F \sqbrk \alpha} F = \index E F = n$

by Tower Law, this implies

$\index E {F \sqbrk \alpha} = 1$

we conclude that $E = F \sqbrk \alpha$.

$\blacksquare$

Also see

• Steinitz's Theorem

Sources

• 1996: Patrick Morandi: Field and Galois Theory: Chapter $1$: Galois Theory: $\S 5$: The Fundamental Theorem of Galois Theory: Corollary $5.7$
• 2002: Serge Lang: Algebra (Revised 3rd ed.): Chapter $\text V}: $\S 4$: Separable Extensions: second part of Theorem $4.6$
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): simple extension