Let $F$ be a finite field such that for every $a \in F$, the equation $x^2=a$ has a solution for $x \in F$ , then what can we say about the number of elements in $F$ and characteristic of $F$?
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Hint: To say that for all $a\in F$, $x^2 = a$ has a solution in $F$ is to say that the squaring map $x\mapsto x^2$ is surjective. Since $F$ is finite every surjective map $F\to F$ is injective. From this, you should be able to determine the characteristic of $F$...
Alex Kruckman
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Yes , true the map is injective , but how do I get information about characteristic form that ? – Souvik Dey Dec 09 '14 at 04:39
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1Recall that $(-1)^2 = 1^2$... – Alex Kruckman Dec 09 '14 at 04:41
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oh yes , thank you – Souvik Dey Dec 09 '14 at 04:42