# In studying abstract algebra, is vector space structure important?

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?

• Short answer: yes! It's very important. – manooooh Sep 8 '18 at 3:08
• Vector spaces are ubiquitous in math. – saulspatz Sep 8 '18 at 3:20
• 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. – Jyrki Lahtonen Sep 8 '18 at 6:01

• 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. – Jonathan Dunay Sep 8 '18 at 3:40
• A field extension $L$ of a field $K$ is a vector space over $K$. – Wuestenfux Sep 8 '18 at 6:01