Norms, discriminants, etc.

Hi,

I have been going though the summary of 3704, and I was wondering if you would be able to give some examples of the methods we use to calculate norms and discriminants as I am having some trouble with it.

I am also not sure how we relate minimal polynomial, norm and trace to linear algebra. Is it that there is a bilinear form?

Thanks for your help

Reply:

Regarding norms and traces, it’s not a bad idea to give yourself some practice just using the definitions. For example, for the field F=Q(\sqrt{2},\sqrt{3}) try to compute the norm and trace of \sqrt{2}+\sqrt{3} using just the definition. First you need to list all the embeddings of F into the complex numbers. How many are there?

But then, there are a number of tricks, like Proposition 70 and Proposition 106. For the discriminants, Proposition 73 is very basic, as well as the basic properties and examples that surround it. Then there are results like Proposition 101, Theorem 102 and Corollary 105. I may be missing a few other points right now, but a flexible usage of those facts will certainly be very basic.

The definition of the discriminant definitely starts from a bilinear form. But in the context I believe you have in mind, the relation to linear algebra I am referring to has to do with the fact that an element of an algebraic number field F can be regarded as a linear map on the field itself. That is, if

\alpha \in F,

then it determines a Q-linear map

L_{\alpha}:F \rightarrow F

by the simple formula

L_{\alpha}(x)=\alpha x

Then the norm of \alpha is equal to the determinant of the linear map L_{\alpha}, while the trace of \alpha is equal to the trace of the linear map L_{\alpha}. Similarly, the minimal polynomial of \alpha is exactly the minimal polynomial of the linear map L_{\alpha}.

This last fact can be useful if \alpha is a generator of F, that is, if F=Q(\alpha). Then the minimal polynomial of \alpha is actually the characteristic polynomial of the linear map L_{\alpha}.

The proof of each of these facts is a rather simple exercise.

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