Exercise 9.20.9. Let $L/K$ be an extension of degree $2$. Show that exactly one of the following happens

the discriminant is $0$, the characteristic of $K$ is $2$, and $L/K$ is purely inseparable obtained by taking a square root of an element of $K$,

the discriminant is $1$, the characteristic of $K$ is $2$, and $L/K$ is separable of degree $2$,

the discriminant is not a square, the characteristic of $K$ is not $2$, and $L$ is obtained from $K$ by taking the square root of the discriminant.

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