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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.

@CoronaCoreanici Maximal subfield of degree a power of 2.

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

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

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

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

Kim Reece@kimreece@mathstodon.xyz[1] http://archive.numdam.org/item/CM_1954-1956__12__35_0/ or https://eudml.org/doc/88821

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