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If $F$ is a finite filed, $\alpha, \beta$ are two non-zero elements of $F$. Then Show that there exists elements $a,b$ in $F$ such that $1+\alpha a^2+\beta b^2=0$.

I don't have any clue on how to approach this. Please help me with this problem.

Tortoise
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    I am not an expert on this topic, but this https://math.stackexchange.com/questions/480509/finite-field-satisfies-1-lambda2-alpha-mu2-0 looks similar to me. – Martin R Nov 04 '17 at 11:14

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The equation is equivalent to $$1+\alpha a^2=-\beta b^2.$$ If more than half the elements in the finite field $F$ are squares, then $1+\alpha a^2$ takes more than half the values in $F$ and $-\beta b^2$ takes more than half the values in $F$. So there has to be a collision: some value of $1+\alpha a^2$ must equal some value of $-\beta b^2$.

If $F$ has even order, every element of $F$ is a square.

If $F$ has odd order, it has $\frac12(1+|F|)$ squares.

Angina Seng
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