I'm studying for an exam and encountered a confusing proof of the following fact in my notes:

Let $[E:F]$ be finite and $\alpha \in E$ then there is an irreducible polynomial $p(x) \in F[x]$ with $p(\alpha) = 0$

My notes start with that $\{1, \alpha, ..., \alpha^{n-1}\}$ were linearly dependent (shouldn't this say independent?) in $E$ over $F$ and defining a nonempty ideal $I = \{p(x) \in F[x] : p(\alpha) = 0 \}$ that is generated by some irreducible polynomial $p_0(x)$

It then assumes $p_0(x)$ is not irreducible and then the proof was supposed to derive a contradiction, but it seems to taper off at that point, so I probably missed something. More specifically, it says $p_0 = p_1p_2$ with the degrees of each of the polynomials on the RHS being less than the degree of $p_0$ then mentioning extension fields and thus contradiction.

Could anyone clarify?

  • 1
    $\begingroup$ Let $n$ be the degree of $E$ over $F$. Then $1,\alpha, \dots, \alpha^{n}$ are linearly dependent. $\endgroup$ – André Nicolas Mar 12 '14 at 18:55
  • $\begingroup$ @AndréNicolas My notes say $\{1, \alpha, ..., \alpha^{n-1}\}$ so either the professor assumed the degree was $n-1$ which would be out of the ordinary or I wrote the wrong thing. Probably the latter. $\endgroup$ – Lost Mar 12 '14 at 18:58


1) Let $n$ be the degree of $E$ over $F$. Then $1,\alpha, \dots, \alpha^{n}$ are linearly dependent over $F$.

2) Let $k$ be the smallest integer such that $1,\alpha,\dots,\alpha^k$ are linearly dependent.

  • $\begingroup$ This theorem showed up on the exam. I was able to prove it (at least I think so) using this hint, so thanks. $\endgroup$ – Lost Mar 12 '14 at 20:50
  • $\begingroup$ You are welcome. Yes, if we pick the least $k$, it is straightforward to show the polynomial we get is irreducible, for if not, we could come up with a smaller $k$. $\endgroup$ – André Nicolas Mar 12 '14 at 20:53

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.