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The ten-field $D_{4}$ correspondence for the splitting field of $x^{4} - 2$

Example

Let $a$ be the positive real fourth root of $2$ , let $i^{2} = - 1$ , and put $L = Q (a, i)$ . Define

$r (a) = i a, r (i) = i, s (a) = a, s (i) = - i .$

Then $Gal (L / Q) = ⟨ r, s : r^{4} = s^{2} = 1, s r s = r^{- 1} ⟩ ≅ D_{4}$ , and the complete correspondence is

Subgroup	Fixed field
${1}$	$Q (a, i)$
$⟨ r^{2} ⟩$	$Q (a^{2}, i)$
$⟨ s ⟩$	$Q (a)$
$⟨ r s ⟩$	$Q (a (1 + i))$
$⟨ r^{2} s ⟩$	$Q (i a)$
$⟨ r^{3} s ⟩$	$Q (a (1 - i))$
$⟨ r ⟩$	$Q (i)$
$⟨ r^{2}, s ⟩$	$Q (a^{2})$
$⟨ r^{2}, r s ⟩$	$Q (i a^{2})$
$D_{4}$	$Q$

Facts & Assumptions

Given: Eisenstein's irreducibility criterion (Eisenstein criterion over the integers) and the normality correspondence of Normal subgroups, conjugate fields, and quotient groups in the Galois correspondence.

[L1]

In the finite Galois correspondence, $[K : K^{H}] = ∣ H ∣$ and $[K^{H} : F] = [G : H]$ , and the subgroup and intermediate-field assignments are mutually inverse bijections (The fundamental theorem of finite Galois theory).

[L2]

For a finite extension $L / E$ with $G = Aut (L / E)$ , being Galois, being the splitting field of a separable polynomial, $∣ G ∣ = [L : E]$ , and $L^{G} = E$ are equivalent (Equivalent characterizations of a finite Galois extension).

Verification

technique · direct

1.1

Eisenstein makes $x^{4} - 2$ irreducible, so $[Q (a) : Q] = 4$ . Since $Q (a) \subseteq R$ and $i \notin R$ , adjoining $i$ doubles the degree. Thus $1, a, a^{2}, a^{3}, i, i a, i a^{2}, i a^{3}$ is a basis of the degree-eight splitting field $L$ .

givenalgebra

2.1

The displayed maps preserve $a^{4} = 2$ and $i^{2} = - 1$ , permute the roots $a, i a, - a, - i a$ , and satisfy $r^{4} = s^{2} = 1$ and $s r s = r^{- 1}$ ; their eight composites are distinct. Those four roots are distinct, so $x^{4} - 2$ is separable and step 1.1 makes $L$ its splitting field; by [L2], $L / Q$ is finite Galois with $∣ Gal (L / Q) ∣ = [L : Q] = 8$ . The eight composites therefore exhaust the automorphism group and give $D_{4}$ .

step 1.1L2construct

3.1

Applying the generators to the eight basis coefficients verifies that every field in the table is fixed by its displayed subgroup. Their degrees over $Q$ are respectively $8, 4, 4, 4, 4, 4, 2, 2, 2, 1$ .

step 2.1algebra

4.1

Those degrees equal the subgroup indices required by [L1], so each containment in step 3.1 is equality and the table is complete. The four reflection subgroups are nonnormal and give the nonnormal quartic fields; the remaining subgroups are normal and give the normal strict fields or endpoints.

step 3.1L1∎

Depends on

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Sources

K. Conrad, The Galois Correspondence, Examples 4.7 and 5.9 (standard reference, not scraped)

The ten-field D4D4​ correspondence for the splitting field of x4−2x4−2