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Let $K=\mathbb{Z}_2 (x,y)$, where $x,y$ are independent, and $L$ be a splitting field extension of $(X^2 - x) (X^2 - y)$, then $[L:K] = 4$ and $L = K(\sqrt{x},\sqrt{y})$ where $\sqrt{x},\sqrt{y}$ are roots of $X^2-x$, $X^2 - y$ respectively. What are the subextensions of $L:K$?

I know all elements in $L$ square to something in $K$, so all the intermediate fields are $K(\sqrt{k})$ for some $k\in K$, but some of them are the same, say $K(\sqrt{x/y}) = K(\sqrt{xy})$...

Note: $\mathbb{Z}_2$ means $\mathbb{Z} / 2\mathbb{Z}$

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  • $\begingroup$ Maybe not an easy task since there are infinitely many. For example, $K_n=K(\sqrt x+y^{n+1/2})$. $\endgroup$
    – user26857
    Oct 14 '13 at 21:43
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    $\begingroup$ Just to be sure, by $\Bbb Z_2$ do you mean $\Bbb Z/2 \Bbb Z$ ? (and not the $2$-adic integers) ? $\endgroup$
    – mercio
    May 21 '14 at 17:47
  • $\begingroup$ @mercio That's right! $\endgroup$
    – user71815
    May 21 '14 at 18:00
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I don't like writing square roots, so let's just pick $K = \Bbb F_2(X^2,Y^2)$ and $L = F_2(X,Y)$. Since $K \subset L$ is of degree $4$, any intermediate field is of degree $2$. As you said, every element of $L$ square to something in $K$ (in fact, the map $x \in L \mapsto x^2 \in K$ is an isomorphism of fields), so those fields are the $K(a)$ for $a \in L \setminus K$ .

The intermediate fields are the $2$-dimensional $K$-vector spaces containing $K$. One direction is obvious, and if $F = \langle 1,a \rangle$ is such a vector space, then $F$ is stable by multiplication (because $1 \cdot a = a \in F$, and $a \cdot a = a^2 \in K \subset F$), so it is the field $K(a)$.

Those correspond to $1$-dimensional vector spaces in the ($3$-dimensional) quotient $L/K$, and so to elements of $(L/K)^* / K^* \simeq \Bbb P^2(K)$. Given an element $[x:y:z] \in \Bbb P^2(K)$ we can associate $a = xX+yY+zXY$ and the intermediate field $K(a)$.

Anyway this means that there are infinitely many (because $K$ is an infinite field) intermediate fields.

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  • $\begingroup$ What does $\Bbb P^2(K)$ stand for? $\endgroup$
    – Ri-Li
    Nov 3 '15 at 20:13
  • $\begingroup$ And will you explain $(L/K)^* / K^* \simeq \Bbb P^2(K)$ this line? $\endgroup$
    – Ri-Li
    Nov 3 '15 at 20:14
  • $\begingroup$ @user152715 P²(K) (the projective plane over K) stands for the set of lines through the origin in a K-vector space of dimension 3, such as (L/K). $\endgroup$
    – mercio
    Nov 3 '15 at 20:58
  • $\begingroup$ 2-dimensional subspaces of L containing K <-> 1-dimensional subspaces (i.e. lines) of L/K <-> elements of P²(K) $\endgroup$
    – mercio
    Nov 3 '15 at 21:01
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    $\begingroup$ @A.P. yes, I'm only talking about $K$-vector spaces that are subsets of $L$, because we are talking about subfields of $L$ here. $\endgroup$
    – mercio
    Nov 4 '15 at 16:18

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