$x,y\in L$ tow algebraic element such that $xy,x+y\in K$. $x,y\in K$?

Question

if $L/K$ be a extension field and $x,y\in L$ tow algebraic element over K such that $xy,x+y\in K$. Can we say that $x,y\in K$?

You can say that $x,y$ belong to some quadratic extension of $K$. — markvs, Jan 29 '22 at 17:37
Note that $x,y$ are the roots of the quadratic $X^2-(x+y)X+xy$ over $K$. — lulu, Jan 29 '22 at 17:54

score 1 · Accepted Answer · answered Jan 29 '22 at 17:36

1

No. Let $K = \mathbf Q$, $L= \mathbf Q[\sqrt 2]$, $x = -y = \sqrt 2$.

answered Jan 29 '22 at 17:36

martini

1 Answers1