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I am looking for an example of a quadratic extension K/F that is not given by $K=F(\sqrt{\beta})$ with $\beta \in F$ and $\beta$ not a square in $F$.

I Know that every quadratic extension (with char(F) $\neq 2$) are of the form $F(\sqrt{\alpha})$, with $\alpha$ not a square in F.

Therefore I assumed $F= \mathbb{Z}/2\mathbb{Z}$. I am not sure what K could be.

usere5225321
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1 Answers1

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The quadratic extension of $\mathbb{F}_2$ is unique up to isomorphism. It can be realised as $\mathbb{F}_2[x]/(x^2 + x + 1)$, that is the polynomials with coefficients in $\mathbb{F}_2$ modulo $x^2 + x + 1$.

Max Freiburghaus
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