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- #1

- Mar 10, 2012

- 835

Now here's my question. Take $p(x)=x^2-4 \in F[x]$, $F$ is the field of Complex Numbers. What is the splitting over $F$ for $p(x)$??

I would be tempted to say that $F$ itself is the splitting field over $F$ for $p(x)$. But then $\mathbb{R}$, the field of real numbers, would be a proper sub-field of $F$ in which $p(x)$ can be factored as a product of linear factors, viz, p(x)=(x-2)(x+2).

What have I missed?