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A minimal generating family in a finite exponent-one purely inseparable extension is a $p$ -basis and gives degree $p^{r}$

Statement

Let $K / F$ be a finite exponent-one purely inseparable extension of characteristic $p$ , and let $(b_{1}, \dots, b_{r})$ be a minimal generating family for $K$ over $F$ . Then it is a $p$ -basis, and

$[K : F] = p^{r} .$

Conversely, every $p$ -basis generates $K$ over $F$ . The empty family gives the trivial extension and degree $p^{0} = 1$ .

Facts & Assumptions

Given: A finite exponent-one purely inseparable extension $K / F$ and a minimal generating family $(b_{1}, \dots, b_{r})$ .

[L1]

If a constant is not a $p$ th power in a characteristic- $p$ field, then $x^{p} - a$ is irreducible (If $a$ is not a $p$ th power in a characteristic- $p$ field, then $x^{p^{n}} - a$ is irreducible for every $n \geq 1$ ).

[L2]

A simple algebraic extension has the power basis whose length is the degree of the minimal polynomial (A simple algebraic extension is its minimal-polynomial quotient and has power basis $1, a, \dots, a^{n - 1}$ and degree $n$ ).

[L3]

Products of bases in a finite tower form a basis of the top field over the bottom field (Products of bases form a basis in a tower of finite extensions).

[L4]

Degrees multiply in a finite tower (Tower law for finite extensions: $[L : F] = [L : K] [K : F]$ ).

[L5]

A $p$ -basis is the restricted-monomial basis of $p$ -bases for finite exponent-one purely inseparable extensions.

Proof

technique · direct

1.1

Put $K_{j} = F (b_{1}, \dots, b_{j})$ . Minimality gives $b_{j} \notin K_{j - 1}$ , while exponent one gives $b_{j}^{p} = a_{j} \in F \subseteq K_{j - 1}$ . If $a_{j} = c^{p}$ for some $c \in K_{j - 1}$ , injectivity of Frobenius in $K$ would give $b_{j} = c$ , a contradiction; hence [L1] makes $x^{p} - a_{j}$ the minimal polynomial of $b_{j}$ over $K_{j - 1}$ .

L1algebra

2.1

By [L2], each step $K_{j} / K_{j - 1}$ has basis $1, b_{j}, \dots, b_{j}^{p - 1}$ and degree $p$ .

step 1.1L2

3.1

Repeated use of [L3] gives the restricted monomials $b_{1}^{e_{1}} \dots b_{r}^{e_{r}}$ as an $F$ -basis of $K$ , so the family is a $p$ -basis by [L5]. Repeated use of [L4] gives $[K : F] = p^{r}$ .

step 2.1L3L4L5

4.1

Conversely, if the restricted monomials form a basis, every element of $K$ is an $F$ -linear combination of products of the $b_{i}$ , so $K = F (b_{1}, \dots, b_{r})$ . For $r = 0$ this says $K = F$ and the degree is one.

L5algebra∎

Depends on

Used by

Fₚ(s,t)/Fₚ(sᵖ,tᵖ) has degree p², infinitely many intermediate fields, and no primitive element Counterexample

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 65 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

P. L. Clark, Field Theory, Chapter 5 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, Chapter 3 (standard reference, not scraped)

A minimal generating family in a finite exponent-one purely inseparable extension is a pp-basis and gives degree prpr