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$⋃_{n \geq 0} F_{p} (t^{1 / p^{n}})$ is an infinite perfect field of characteristic $p$

Example

Inside an algebraic closure of $F_{p} (t)$ , let $t_{n}$ be the unique root of $x^{p^{n}} - t$ and put $F_{n} = F_{p} (t_{n})$ . Then

$P : = ⋃_{n \geq 0} F_{n}$

is an infinite perfect field of characteristic $p$ containing $F_{p} (t)$ .

Facts & Assumptions

Given: A prime $p$ and an algebraic closure $Ω$ of $F_{p} (t)$ .

[L1]

In characteristic $p$ , a field is perfect exactly when Frobenius is surjective (A field is perfect exactly when it has characteristic zero or its Frobenius map is surjective).

[L2]

Frobenius is injective and respects field operations in characteristic $p$ (Frobenius $x \mapsto x^{p}$ is an injective endomorphism in characteristic $p$ , and an automorphism for finite fields).

[L3]

The rational function field is the fraction field of its polynomial ring (For a field $F$ , $F (t) = Frac (F [t])$ is its rational function field; in particular $R (t) = Frac (R [t])$ ).

Verification

technique · direct

1.1

The given algebraic closure contains a root $t_{n}$ of every $x^{p^{n}} - t$ , and [L2] makes that root unique. Take $t_{0} = t$ . Uniqueness gives $t_{n + 1}^{p} = t_{n}$ , so $F_{n} \subseteq F_{n + 1}$ .

L2construct

2.1

A nested union of fields is a field, so $P$ is a field of characteristic $p$ containing $F_{0} = F_{p} (t)$ . The distinct powers of the indeterminate $t$ show that this fraction field, and hence $P$ , is infinite.

step 1.1L3algebra

2.2

If $z = r (t_{n}) / s (t_{n}) \in F_{n}$ , replace $t_{n}$ by $t_{n + 1}$ in the same rational expression. Since coefficients in $F_{p}$ are fixed by Frobenius, [L2] shows that the resulting element of $F_{n + 1}$ has $p$ th power $z$ . Thus Frobenius on $P$ is surjective.

step 1.1L2algebra

3.1

By [L1], $P$ is perfect.

step 2.2L1∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 60 results over 15 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 4 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, Chapters 3 and 5 (standard reference, not scraped)

⋃n≥0Fp(t1/pn)⋃n≥0​Fp​(t1/pn) is an infinite perfect field of characteristic pp