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Let $\ell$ be a skew field (i.e., a division ring), and let $k$ be a sub skew field, such that the dimension of $\ell$ as a left vector space over $k$ is $2$.

Then if $a \in \ell \setminus k$, we can write each element $b \in \ell$ as $b = b_1 + b_1a$, with $b_1, b_2 \in k$.

Question: does there exists an element $c \in \ell \setminus k$ such that we can write $c^2$ as $c^2 = c_1 + cc_2$, with $c_1, c_2 \in k$ ? (Here, we allow that $c_2 = 0$.)

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  • $\begingroup$ I don't think I understand the question; it seems to me that the positive answer follows from what you have written, for every $c \in \ell \setminus k$, by putting $a = c$ and $b = c^2$. $\endgroup$
    – LSpice
    Commented Jun 13 at 17:24
  • $\begingroup$ My guess for what you want to ask: does every element of of $\ell \setminus k$ satisfy a quadratic equation over $k$? $\endgroup$
    – Kimball
    Commented Jun 14 at 6:02
  • $\begingroup$ @Kimball: we know that already: since $\ell$ is a left vector space of dimension 2 over $k$, we can write $c^2$ (for each $c$ not in $k$) as $a_1 + a_2c$ for some $a_1, a_2 \in k$. The question is: can we find some $c$ not in $k$ so that $c^2 = c_1 + cc_2$ ? $\endgroup$
    – THC
    Commented Jun 14 at 12:23
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    $\begingroup$ @Kimball: we do not know the dimension of $\ell$ as a right vector space over $k$ ... $\endgroup$
    – THC
    Commented Jun 20 at 6:47
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    $\begingroup$ If the division rings are finite-dimensional over their center, then of course the left and right dimensions of $\ell$ over $k$ coincide. But indeed, for all finite $m,n > 1$, there exist examples of skew field extensions $\ell$ over $k$ such that the left dimension is $m$ and the right dimension is $n$. (See mathoverflow.net/questions/320526/…) $\endgroup$
    – Tom De Medts
    Commented Jun 21 at 8:00

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