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Let $K\supseteq F$ be a field extension and $x,y\in F$. Can someone please explain to why the equivalence $$ x,y\text{ algebraic over }F\ \Leftrightarrow\ x+y\text{ and }xy\text{ are algebraic over }F $$ should hold ?

(the $\Rightarrow$ direction is fairly easy, and I did it, but I'm stuck at the other direction.)

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Actually, it is $\implies$ direction that requires some work, certainly more than the other direction. – André Nicolas Oct 30 '12 at 19:09
up vote 4 down vote accepted

$x$ and $y$ both satisfy the equation $X^2-(x+y)X+xy$, so they are algebraic over $F(xy, x+y)$. Now recall that if $F\subset K \subset L $ and $L/K$ and $K/F$ are algebraic, so is $L/K$. (You can prove this by using an argument about dimension, since every algebraic element is contained in a finite extension.)

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You meant $X^2 -(x+y)X+xy$ – Lubin Oct 30 '12 at 19:37
Indeed I did. Just fixed it. – Brett Frankel Oct 30 '12 at 20:54

Look at the roots of $$\alpha^2 - (x+y) \alpha + xy$$

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