en.wikipedia.org/wiki/Fundamental_theorem_of_Galois_theory
1 correction found
The only intermediate field that contains F = Q is E = Q ( 2 3 ) .
This is incorrect because Q itself is also an intermediate field of Q(∛2)/Q. In fact, for this degree-3 extension the intermediate fields are exactly Q and Q(∛2).
Full reasoning
An intermediate field (or subextension) of an extension E/F is, by definition, a subfield of E that contains F. So F itself always counts as an intermediate field.
Here E = Q(∛2) and F = Q, so Q is an intermediate field of E/F. That alone makes the sentence false: E is not the only intermediate field containing F.
More specifically, Q(∛2) has degree 3 over Q. For any intermediate field K, multiplicativity of degrees gives
[E:Q] = [E:K][K:Q].
Since [E:Q] = 3 is prime, the only possibilities are [K:Q] = 1 or 3, so K = Q or K = E. Thus this example has no proper intermediate fields, but it still has two intermediate fields: Q and Q(∛2).
So the sentence should say something like: there is no proper intermediate field between Q and Q(∛2), not that E is the only intermediate field containing F.
3 sources
- Section 9.6 (09FT): Field extensions — The Stacks Project
Let F/k be a field extension. Let S ⊂ F be any subset. Then there is a smallest subextension of F (that is, a subfield of F containing k) that contains S.
- Lemma 9.7.7 (09G9): Multiplicativity — The Stacks Project
Suppose given a tower of fields F/E/k. Then [F:k] = [F:E][E:k].
- 4000/6000 Day 24: Dimension of vector spaces
Corollary: The field Q(sqrt(2)) has dimension 2 over Q, and Q(cuberoot(2)) has dimension 3 over Q.