0
$\begingroup$

Suppose $L$ is an extension field of $K$ and $L$ is algebraically closed. Then is it trivially true that $\forall f(x)\in K[x],~f(x)$ splits over $L$? I think the answer is yes since $L$ is algebraically closed means that every polynomial with coefficients in $L$ have a root in $L$ (and hence have all roots in $L$)

$\endgroup$
  • 1
    $\begingroup$ Yes, this follows by induction on the degree of $f$, starting with one root. So there is nothing to ask in your question. $\endgroup$ – Dietrich Burde Dec 15 '18 at 12:37
0
$\begingroup$

Yes, the splitting field of $f(x)\in K[x]$ must be contained in $L$ since $L$ is algebraically closed.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.