I'm new to algebraic number theory and field extension theory. From what I've understood, a finite algebraic field extension $L/K$ is a vector space over $K$ of dimension $n$ and can be seen as $K[x]/(p(x))$ for $p$ an irreducible polynomial of degree $n$.

I've read that a polynomial is said separable on $K$ if it has distinct roots, that an element in $L/K$ is separable if its minimal polynomial is separable and that $L/K$ is said separable if all its elements are separable.

Now, in Ireland and Rosen, chapter 12, it is said that a separable field extension $L/K$ of degree $n$ possesses (exactly) $n$ distinct isomorphisms $\sigma_1,\dots,\sigma_n$ into a finite algebraic closure of $K$ that leave $K$ fixed. One of these isomorphisms, usually $\sigma_1$, is the identity.

For $L/K=\Bbb Q(i)\cong Q[x]/(x^2+1)$ I know from class that $\sigma_1:a+bi\mapsto a+bi$ and $\sigma_2:a+bi\mapsto a-bi$ but I don't know how to identify these isomorphisms in general.

Could you help me? For instance, what are the 3 isomorphisms that leave $\Bbb Q$ fixed in $F=\Bbb Q(\theta)$ where $\theta$ is a root of the irreducible polynomial $x^3-4x+2$?

  • 2
    $\begingroup$ It just picks out the various roots of $p$. $\endgroup$ – Alex Youcis May 4 '14 at 14:16

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