Welcome to MathOverflow

Question

2

Let $L/K$ be a field extension. Let $\alpha \in L$ be algebraic over $K$. Is there an established terminology for the property of $\alpha$ that $K(\alpha)/K$ is a normal field extension? Would you call $\alpha$ normal over $K$ in case $K(\alpha)/K$ is normal?

edited Nov 22 2010 at 3:14

Allen Knutson
12.4k●1●23●64

asked Nov 20 2010 at 17:30

Marc Nieper-Wißkirchen
1,313●5●15

2

I have been studying field theory for some time, and I have never encountered any terminology for the situation you describe. My first instinct was that calling such an element "normal" is not such a great idea, but I find that I can't defend that at all: upon reflection, it seems reasonable enough. – Pete L. Clark Nov 20 2010 at 18:06

1

@Marc: You can't avoid field extensions systematically in such questions, since at some point you have to compare different extensions of a given field which are isomorphic. Elements outside the ground field have to live somewhere definite. – Jim Humphreys Nov 20 2010 at 18:24

1

@Marc: Not to prolong the dialogue too much, for me an algebraic subject like Galois theory is better treated algebraically without presupposing knowledge of complex numbers and such. In any case, Galois theory requires systematic study of isomorphisms and automorphisms for finite field extensions: consider the Galois group. Galois himself had the insight for this but lacked the language to make it transparent. – Jim Humphreys Nov 20 2010 at 19:41

3

The phrase "normal element" seems to be taken already, too, as it is used in the literature to refer to an element which, with its conjugates, forms a basis (called a "normal basis") for the extension. In that sense, $i$ is not a normal element of ${\bf Q}(i)/{\bf Q}$, since $\lbrace i,-i\rbrace$ is not a basis. – Gerry Myerson Nov 21 2010 at 1:02

2

I agree that "normal element" is bad, primarily because "normal extension" isn't that terrific in the first place. "Saturated extension" would be better, perhaps. For an element, if you really need terminology, let me suggest "gregarious" or "fraternal" since it cannot leave its conjugates. – Laurent Moret-Bailly Nov 22 2010 at 7:45

show 9 more comments

What do you call an algebraic element with the property that the generated field extension is normal?

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.