# Slick and fast linear algebra treatment for finite field extensions?

I am looking for a quick introduction to linear algebra that is

• slick (i.e., proofs are easy to follow and don't require handwaving);

• general and comprehensive enough to serve as foundation for finite field extensions (i.e., it works over any field, and does arbitrary vector spaces and not just $$F^n$$);

• reasonably short;

• and yet reasonably elementary that students who have mostly seen matrix computations and $$\mathbb{R}^n$$ will recognize ideas.

Background: I am teaching an abstract algebra class to an audience of rather diverse skillsets. A significant number of the students will be bored by a lengthy from-scratch review of linear algebra, while another will be lost without at least some kind of remediation. The material should include whatever is necessary for the theory of finite field extensions (up to basic Galois theory) -- so, quotient spaces, direct sums, kernels at least.

I don't need determinants (I will do them anyway, following Strickland, so they can be assumed), infinite-dimensional vector spaces (except for their existence and the fact that $$F\left[x\right]$$ is an $$F$$-vector space with basis $$\left(1,x,x^2,\ldots\right)$$), diagonalization, nilpotency, orthogonality, bilinear forms, numerics, symmetric and alternating matrices, inequalities.

I don't mind if the proofs are terse, assuming that they can be expanded without too much trouble. What I care about is that the results are arranged in a way to make short proofs possible.

• What about Serge Lang's Linear Algebra? – Dietrich Burde Mar 4 at 21:44
• @DietrichBurde: I see no quotient spaces. Chapters III and IV seem otherwise pretty good, but I'm worried about further holes in coverage that will reveal when I try some Galois basics. – darij grinberg Mar 5 at 2:02
• For possible holes I would add another classical reference, like Kowalski, which is too long otherwise. – Dietrich Burde Mar 5 at 9:12

A Book Of Abstract Algebra by Pinter has a chapter that's a quick overview of linear algebra for finite field extensions. It's inexpensive, though possibly not as comprehensive as you need.

• Hmm. I'm not seeing quotient spaces in there, but thanks for the p(o)inter. – darij grinberg Mar 5 at 2:17

Anthony W. Knapp, Basic Algebra (Digital Second Edition, 2016) is freely downloadable from the link just given, or from Project Euclid; so bulk and cost are not an issue, even though the book also covers many other topics.

However, if it is necessary to print any part of it:

Printing of the files is constrained legally. The reason is that Springer Science+Business Media, Inc. has a stake in printed copies of these books, as is explained in the brief history below, and Springer's permission may be required for printing.

I haven't yet read it myself, but it seems to be well worth a look.

The topic of quotients of vector spaces is treated in section 5 of Chapter II. The restriction to $$\mathbb{Q}$$, $$\mathbb{R}$$ and $$\mathbb{C}$$ in the chapter title does not seem to matter.

The topic of finite fields is treated in section 3 of Chapter IX.

• Thank you! I was aware of the Knapp books, but I didn't think of looking into them for linear algebra (I thought they started above it). The treatment looks like it does quotients really nicely. – darij grinberg Mar 5 at 16:09