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$Q (ζ_{3}, \sqrt[3]{2}, \sqrt[3]{3})$ is a Kummer extension with quotient $(Z / 3)^{2}$

Example

Let $F = Q (ζ_{3})$ and

$K = F (\sqrt[3]{2}, \sqrt[3]{3}) .$

Then $K / F$ is a Kummer extension, and the subgroup generated by the classes of $2$ and $3$ in $F^{\times} / (F^{\times})^{3}$ is isomorphic to $(Z / 3)^{2}$ . Consequently

$Gal (K / F) ≅ (Z / 3)^{2} .$

Facts & Assumptions

Given: The field $F = Q (ζ_{3})$ and the extension $K = F (\sqrt[3]{2}, \sqrt[3]{3})$ .

[L1]

Kummer theory identifies finite abelian extensions of exponent dividing $3$ with subgroups of $F^{\times} / (F^{\times})^{3}$ (Kummer theory classifies finite abelian extensions of exponent dividing $n$ by subgroups between $(F^{\times})^{n}$ and $F^{\times}$ ).

[L2]

Norm and trace are given by the embedding formulas (Norm and trace from embeddings, with the inseparable exponent in the norm formula).

Verification

technique · direct

1.1

The field $F$ contains $μ_{3}$ , so adjoining cube roots is exactly the Kummer situation of [L1]. To show that the classes of $2$ and $3$ are independent modulo cubes, suppose $2^{a} 3^{b} = c^{3} in F^{\times} .$ Taking norms from $F$ to $Q$ gives $2^{2 a} 3^{2 b} = N_{F / Q} (c)^{3} .$ The left side is a rational cube only when $3 ∣ 2 a$ and $3 ∣ 2 b$ , hence only when $3 ∣ a$ and $3 ∣ b$ . Therefore the classes of $2$ and $3$ each have order $3$ and generate a subgroup isomorphic to $(Z / 3)^{2}$ .

L2algebra

2.1

By [L1], the field generated by the corresponding cube roots is a Kummer extension with Galois group $Hom ((Z / 3)^{2}, μ_{3}) ≅ (Z / 3)^{2} .$ Since $K$ is exactly that field, the stated conclusion follows.

L1step 1.1∎

Depends on

Used by

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Dependency tree · two levels

18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

B. Conrad, Kummer Theory, Theorem 5.12 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, v5.10, Theorem 5.30 (standard reference, not scraped)

Q(ζ3,23,33)Q(ζ3​,32​,33​) is a Kummer extension with quotient (Z/3)2(Z/3)2