Exercise 5.1.1

Show that a splitting field of x 3 − 2 over ℚ is ℚ ( ω , 2 3 ) , ω = e 2 πi ∕ 3 .

Answers

Proof. The roots of x 3 − 2 are 2 3 , ω 2 3 , ω 2 2 3 . A splitting field of x 3 − 2 over ℚ is thus ℚ ( 2 3 , ω 2 3 , ω 2 2 3 ) ⊂ ℂ .

As ω , 2 3 ∈ ℚ ( ω , 2 3 ) , and as ℚ ( ω , 2 3 ) is a field, 2 3 , ω 2 3 , ω 2 2 3 are elements of ℚ ( ω , 2 3 ) . Since ℚ ( 2 3 , ω 2 3 , ω 2 2 3 ) is the smallest subfield of ℂ containing ℚ and 2 3 , ω 2 3 , ω 2 2 3 ,

ℚ ( 2 3 , ω 2 3 , ω 2 2 3 ) ⊂ ℚ ( ω , 2 3 ) .

Moreover ω = ω 2 3 ∕ 2 3 ∈ ℚ ( 2 3 , ω 2 3 , ω 2 2 3 ) and 2 3 ∈ ℚ ( 2 3 , ω 2 3 , ω 2 2 3 ) . As ℚ ( ω , 2 3 ) is the smallest subfield of ℂ containing these two elements,

ℚ ( ω , 2 3 ) ⊂ ℚ ( 2 3 , ω 2 3 , ω 2 2 3 ) .

These two subfields are identical.

Conclusion : a splitting field of x 3 − 2 over ℚ is ℚ ( ω , 2 3 ) . □

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2022-07-19 00:00
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