# Order considerations of a finite field with square roots for every element

Let $\mathbb{F}$ be a finite field such that for every $a ∈ \mathbb{F}$ the equation $x^2 =a$ has a solution in $\mathbb{F}$. Then which is true?
1. The characteristic of $\mathbb{F}$ must be $2$
2. $\mathbb{F}$ must have a square number of elements
3. The order of $\mathbb{F}$ is a power of $3$
4. $\mathbb{F}$ must be a field with prime number of elements

Can anyone suggest to me how I can solve this problem?

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See whether it's true in the field of $8$ elements --- that could answer several questions in one go. –  Gerry Myerson Dec 18 '12 at 5:31
i did not understand your hint. can you explain please.thank you. –  pinti Dec 18 '12 at 5:54
I assume you are aware that there is a field of $8$ elements (if not, then you have not reached the point in your studies where you can raise these questions). If you can show $x^2=a$ always has solutions in this field, do you not see the implications for questions 2, 3, and 4? –  Gerry Myerson Dec 18 '12 at 5:59

Hint: The equation $x^2=1$ has $1$ and $-1$ as solutions. Under what conditions are these the same?