I was given a problem that I don't seem to know how to solve. It says

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> Let $\mathbb{F}$ be a field with $7$ elements. Construct a field $\mathbb{L}$ with $7^2$ elements and show that $x^2-3$ and $x^2+x-1$ can be solved in $\mathbb{L}$. Now construct a field $\mathbb{K}$ with $7^3$ elements and show that $x^2-3$ cannot be solved in $\mathbb{K}$.
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So far, I think that $\mathbb{F}$ must be $\mathbb{F}\_7=\mathbb{Z}/7\mathbb{Z}$ and, given that $49$ is not prime, $\mathbb{L}$ must be $\mathbb{L}=\mathbb{F}\_7[x]/(f(x))$ but I am having trouble finding $f(x)$. I was thinking that $f(x)=x^2-5$ would work since $\sqrt{5}\notin\mathbb{F}\_7$ and $\sqrt{5}$ is needed to solve both equations. However, if I use $x^2-5$, I can't seem to solve for $\sqrt{5}$. Also, I have no idea what to do for $\mathbb{K}$.

Thanks