I need help finding the hole in this argument...

Let K be a CM number field, K+ its maximal real subfield, k the 2-part of K with subfield k+ likewise. Suppose K has a purely imaginary unit a. Then by Remak [1], a is of the form \sqrt{-u} for a totally positive non-square unit u of K+. The degree of K over k is odd, therefore the norm N_{K/k}(a) is also purely imaginary, and a unit. Therefore a totally positive non-square unit exists in k+, and moreover it is found similarly.

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Note: I have in my possession a counter-example.

I forgot to add, i is not in K

@kimreece I'd help but I don't even know what a 2-part is defined as, so unfortunately I doubt I can help much.

@CoronaCoreanici Maximal subfield of degree a power of 2.

@CoronaCoreanici Honestly... while I would really really like an explanation of how this works. I'm miserable about it. Because there's definitely something wrong. I have, in my possession, a unit which is totally positive real non-square and thus can be used to generate such a CM field, yet its norm by the odd part of the field is .. a square in the field below. f.f.f.f.f. How. the. f. The only thing I can imagine is that I'm somehow misreading Remak. And that, well, I need a German reader for

@kimreece Yeah, can't do you for there. I was reading through your proof sketch blackboxing that.

@CoronaCoreanici purely imaginary unit, take odd degree partial norm, I should get a purely imaginary unit, right? That's, like, kindergarten multiplication..

@CoronaCoreanici And I can't have sqrt{-1} below without having it above; that's set containment.

@CoronaCoreanici It has to be something either in Remak or in my reading of it, doesn't it? I mean, is there anywhere else it could go wrong?

@kimreece Possibly? I don't know how that norm works.

@CoronaCoreanici It's just the product over the conjugate embeddings. Which, for a purely imaginary number in a CM field, should all be purely imaginary.

@kimreece Are you sure you still get a unit in that case?

@CoronaCoreanici Good question... but yes. Partial norms of units are units. They still have the same absolute norm of +- 1 as before, and they haven't ceased to be algebraic integers.

@CoronaCoreanici Actually it's worse than that; I have by Garbanati (and his proof looks really good) that the field below in that case /can't/ have a real totally positive non-square unit... 🤦‍♂️

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