Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $L/K$ a finite extension and $f(x)\in K[x]$ a non-linear irreducible polynomial. Prove that if $\mathrm{gcd}\left( \mathrm{deg}(f) , \left[ L:K \right] \right)=1$ then $f(x)$ has no roots in $L$.

Added: (Solution based on the answer below)

Suppose $f(x)$ has a root in $L$, namely $\alpha$ and consider the extension $K(\alpha)/K$. Since $f$ is irreducible we have that $[K(\alpha) : K] = \mathrm{deg}(f) > 1$. On the other hand we have that $[L:K]=[L:K(\alpha)][K(\alpha):K]$. Then $[L:K]=[L:K(\alpha)](\mathrm{deg}(f))$ but this is imposible since $\mathrm{gcd}\left( \mathrm{deg}(f) , \left[ L:K \right] \right)=1$ and $\mathrm{deg}(f) > 1$.

share|cite|improve this question
I would just add "and $\mathrm{deg}(f)\gt 1$" at the end; otherwise, perfect. (It my even be okay without it; I just like hammering the nails in solidly). – Arturo Magidin Jan 12 '11 at 4:14
up vote 8 down vote accepted
  1. What do you know about the degree $[K(\alpha):K]$ of an extension when $\alpha$ is a root of an irreducible polynomial $g(x)\in K[x]$?

  2. What do you know about the degrees $[L:K]$, $[K:F]$, and $[L:F]$ of extensions when you have a tower $F\subset K\subset L$ ?

share|cite|improve this answer
WooW! I guess I was complicating a lot this problem. Thank a lot for your time. And sorry for the dumb question. – Chu Jan 12 '11 at 3:57
@Chu: Don't forget to formally accept an answer (the one you find most helpful) when you are satisfied. I might also suggest that you edit your question to add your solution if you with; that way we can help you with that too. – Arturo Magidin Jan 12 '11 at 4:00
Ok. Do i post the solution within the space where the question go? – Chu Jan 12 '11 at 4:02
@Chu: You should have an edit button for the question. I would suggest editing it, and adding the solution at the bottom; perhaps divide it with a horizontal line <hr/> from the original text, mark it as an addition (say, by adding *Edit:* or *Added:*) and then write your solution. Or you can add your solution as an Answer if you prefer. – Arturo Magidin Jan 12 '11 at 4:05
Thank you so much for your time. I'll do that. :) – Chu Jan 12 '11 at 4:06

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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