The school year has started, and I've been sitting in on some graduate-level abstract algebra lectures, without having an algebra background - my background is in applied math, so, lots of analysis, linear algebra, complex variables, probability and numerical methods.

I noticed the algebraic structures that are defined and then used throughout the number-theoretic lectures so far are groups, rings, and fields; however, there has been no mention of vector spaces.

Is vector space structure important, when studying abstract algebra as a first (graduate level) course? Does it ever come up later on, say, in research level questions?

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    $\begingroup$ Short answer: yes! It's very important. $\endgroup$ – manooooh Sep 8 '18 at 3:08
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    $\begingroup$ Vector spaces are ubiquitous in math. $\endgroup$ – saulspatz Sep 8 '18 at 3:20
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    $\begingroup$ Vector spaces are not mentioned in the early parts of that course because everybody was expected to become familiar with them in linear algebra. At this point it is more important to bring in new actors to the play. Trust me, they will be used further down. $\endgroup$ – Jyrki Lahtonen Sep 8 '18 at 6:01

As mentioned in the comments, the short answer is "yes."

In your first course, you don't talk about them as much, probably partly because there are classes that are specifically linear algebra classes. However, it does show up in abstract algebra. Here are a few places that it shows up:

  • When studying field extensions and Galois theory, linear algebra shows up in the proofs of fundamental results within this topic.
  • Modules are actually a generalization of vector spaces (they have the same properties except that they are over a ring rather than over a field)
    • As a remark, if you have results from modules over a PID several linear algebra results follow immediately.
  • In representation theory, objects like groups act on vector spaces
  • A Lie algebra is a vector space with additional structure
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    $\begingroup$ So, for group representations, you are defining a product $G \times V \to V$ which in short is compatible with both the group structure and vector space structure. Equivalently, you have a homomorphism $\rho:G \to GL(V)$. There are other types of representations as well. Representation Theory by Fulton and Harris is a good book for this topic. $\endgroup$ – Jonathan Dunay Sep 8 '18 at 3:40
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    $\begingroup$ Although don't start in on this book unless you are already comfortable to some extent with abstract algebra $\endgroup$ – Jonathan Dunay Sep 8 '18 at 3:40
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    $\begingroup$ A field extension $L$ of a field $K$ is a vector space over $K$. $\endgroup$ – Wuestenfux Sep 8 '18 at 6:01

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