Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

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

An algebraic field extension $K \subset E$ is called normal if $E$ is the splitting field of a collection of polynomials with coefficients in $K$.

So if $K \subset E$ is some field extension and $E$ is the splitting field of a polynomial $f(x) \in K[x]$, then this extension is normal, because i can take my family of polynomials to be this single polynomial $f(x)$.

Now let $K \subset E$ be a proper but otherwise arbitrary field extension. Let $\alpha \in K$. Then $E$ is the splitting field of $x - \alpha$ and so $K \subset E$ is normal. But this can not be true because $K \subset E$ was arbitrary.

Where is the flaw in the above line of thought? What am i missing in the definition of a normal extension?


share|cite|improve this question
up vote 3 down vote accepted

$E$ is not the splitting field of $X-\alpha$. You are misunderstanding the definition of a splitting field. A splitting field of a polynomial $f(X)\in K[X]$ is NOT just a field where your polynomial $f(X)$ splits. It's a field extension $E/K$ s.t. $E$ is generated by roots of $f(X)$. In this case $K[\alpha]=K$, so $E$ is not generated by roots of $X-\alpha$ unless $E=K$.

share|cite|improve this answer
That is very enlightening. Thanks. – Manos Dec 4 '11 at 20:35

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.