Question
Let $f(x) \in F[x]$. If $\operatorname{deg} f(x)=2$ and $a$ is a zero of $f(x)$ in some extension of $F$, prove that $F(a)$ is the splitting field for $f(x)$ over $F$.
Step 1
Since $\operatorname{deg} f(x) = 2$, we know that $f(x)$ is a quadratic polynomial. Let $f(x) = ax^2 + bx + c$ with $a, b, c \in F$ and $a \neq 0$. Show more…
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