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Part CCSIR NET December 2023

Galois theory essentials

Let $f (X) = X^{3} - 2 \in Q [X]$ and let $K \subset C$ be the splitting field of f(X) over $Q$ . Let $ω =$ $e^{2 π i /3}$ . Which of the following statements are true?

A.The Galois group of K over $Q$ is the symmetric group $S_{3}$ .✓
B.The Galois group of K over $Q (ω)$ is the symmetric group $S_{3}$ .
C.The Galois group of K over $Q$ is $Z /3 Z$ .
D.The Galois group of K over $Q (ω)$ is $Z /3 Z$ .✓

Solution

$K = Q (32, ω)$ has degree 6 over $Q$ with non-abelian Galois group $S_{3}$ ; over $Q (ω)$ the degree is 3, so the group is $Z /3$ .

Related counterexample: Every extension of degree n has a Galois group of order n

From Rings and Fields › Galois theory essentials

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