When are nonintersecting finite degree field extensions linearly disjoint?

Let F be a field, and let K,L be finite degree field extensions of F inside a common algebraic closure. Consider the following two properties:

(i) K and L are linearly disjoint over F: the natural map KFLKL is an injection.

(ii) KL=F.

It is well known that (i) (ii): see e.g. § 13.1 of my field theory notes. This implication ought to be (and usually is) followed up with the comment that (ii) does not imply (i) without some additional hypothesis: for instance take F=Q, K=Q(32), L=Q(e2πi332), or more generally, any two distinct, but conjugate, extensions of prime degree. Thus some normality hypothesis is necessary. What is the weakest such hypothesis?

The following is a standard result: see e.g. § 13.3, loc. cit.

Theorem: If K/F and L/F are both Galois, then (ii) (i).

I remember this point coming up in a course I took as a graduate student, and the instructor claimed in passing that it was enough for only one of K, L to be Galois. None of the standard field theory texts I own contains a proof of this. But by online searching I found an algebra text of P.M. Cohn which proves something stronger.

Theorem: If at least one of K,L is normal and at least one is separable [it is permissible for the same field to be both normal and separable], then (ii) (i).

I wasn’t able to freely view the proof, so if someone can pass it along to me I’d be appreciative. Still, I think I know of no counterexamples to the following stronger

Claim: If at least one of K,L is normal, then (ii) (i).

Is this in fact true? (I believe I have seen it claimed in certain research papers, e.g. one by Piatetski-Shapiro. But because terminology and running separability hypotheses are not so standardized, I don’t take this as conclusive evidence.)

Answer

There are counterexamples to the Claim. Recall that an extension of a field F of characteristic p>0 is separable if and only if it is linearly disjoint from F1/p over F. Note that F1/p/F is a normal extension. There is an insparable algebraic extension K/F which does not intersect F1/pF (see Does an inseparable extension have a purely inseparable element?). This gives a counterexample.

Attribution
Source : Link , Question Author : Pete L. Clark , Answer Author : ll_n

Leave a Comment