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Why are automorphisms of ℚ(√2) determined by actions on √2? Why is Gal(ℚ(√2)/ℚ) isomorphic to ℤ2?

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Gal(ℚ(ω,∛2)/ℚ) ≈ S3, where ω is a complex cube root of unity: ω = -1/2+i√3/2 = e^(2πi/3)

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Field Extension & Splitting Field Examples | Adjoin Roots: ℚ(√2), ℚ(√3), ℚ(√2,√3)=ℚ(√2)(√3)

Why is the Galois group of ℚ(∛2) over ℚ trivial?

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