Abstract Algebra – Dedekind’s Theorem in Lang’s Algebra

abstract-algebraalgebraic-groupsterminology

This is probably a quick, easy question to answer. I'm having trouble searching for it due to Dedekind's prolific nature.

In Springer's book, "Linear Algebraic Groups (Second Edition)", page 43, $\S$3.2.1, the following is stated:

Let $G$ be a linear algebraic group. A homomorphism of algebraic groups $\chi:G\to\Bbb G_m$ is called a rational character (or simply a character). The set of rational characters is denoted by $X^*(G)$. It has a natural structure of abelian group, which we write additively. The characters are regular functions on $G$, so lie in $k[G]$. By Dedekind's theorem [La2, Ch. VIII, $\S$4] the characters are linearly independent elements of $k[G]$.

Here [La2, Ch. VIII, $\S$4] refers to

S. Lang, Algebra, Addison-Wesley, 1977

Chapter VIII, $\S$4.

However, upon looking through that section of Lang's book, I cannot find any reference to Dedekind; maybe I'm not looking closely enough or something – I don't know.

Please would someone tell me which theorem is meant?

Context:

This is a question and so there isn't much I can add in terms of context. Nevertheless, I will answer the questions listed here as a suitable alternative:

  • What are you studying?

A postgraduate research degree in linear algebraic groups. I'm in my first year.

  • What text is this drawn from, if any? If not, how did the question arise?

See above.

  • What kind of approaches (to similar problems) are you familiar with?

This is not really a problem that has related questions; however, I can say that I have been working on linear algebraic groups since September 2022. I have read & (mostly) understood Chapters 1 and 2 of Springer's book (amongst chapters of other books, like Borel's and Humphreys', and someone's PhD thesis in the area), under the guidance of my academic supervisor (who is on holiday at the moment, hence why I am not asking him).

  • What kind of answer are you looking for? Basic approach, hint, explanation, something else?

A statement of the theorem in question by Dedekind.

  • Is this question something you think you should be able to answer? Why or why not?

Yes, given more time. It seems like a matter of looking things up, but, like I said above, Dedekind was prolific and there's too much out there.

Best Answer

It is literally what he says: different characters of a group are linearly independent. Don't worry about the name Dedekind. Just look for a theorem on linear independence of characters. Search for it online with obvious search terms if you don't find what you want in Lang.

Related Question