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The bookUnit 2 · Rings and Fields48 / 83

Galois theory essentials

Why this is asked: For an exam you need the Galois groups of x³ − 2, x⁴ − 2, xⁿ − 1 and the fundamental correspondence — subgroups ↔ intermediate fields, with normal subgroups ↔ normal extensions.

Galois correspondence and the standard groups

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See it move

The Galois correspondence, used on one exampleinteractive

A degree-4 extension, its group, and the subfield lattice read straight off the subgroup lattice.

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The trap here

“Every extension of degree n has a Galois group of order n” — false

Q(23)/Q\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q} has degree 3 but only the identity automorphism

The extension is not normal — the other cube roots are not real. Order = degree only for Galois extensions.

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Check yourself — select all that apply

Let f(X)=X3−2∈Q[X]f(X) = X^{3} - 2 \in \mathbb{Q}[X] and let K⊂CK \subset \mathbb{C} be the splitting field of f(X) over Q\mathbb{Q}. Let ω=\omega = e2πi/3e^{2\pi{}i/3}. Which of the following statements are true?

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Open this in the full syllabus view · Unit 2