# What is the importance of the characteristic of a ring or field?

I am studying group theory and many of my textbooks mention the characteristic of a field or ring. I understand the definition, however I am having trouble finding examples of what this concept is used for, or it's theoretical meaning.

• It tells us a few things about the ring/field. All fields have a prime (or 0) characteristic, which is not obvious from a field's construction, and reveals a few things about ideals on rings. – Kaynex Apr 23 '17 at 22:13
• Fields of characteristic zero look a bit like (and contain) $\mathbb{Q}$, while fields of positive characteristic behave more like (and contain) a finite field $\mathbb{F}_p$. The fields $\mathbb{Q}, \mathbb{F}_p$ have very different properties. For example, in characteristic $2$ we have $a-b = a+b$, so that there's not really a notion of subtraction. You will see many theorems that assume a field has a given characteristic. Representation theory is much harder in nonzero characteristic, etc. etc. – Jair Taylor Apr 23 '17 at 22:52
• Ah ok, thank you for the response. I'm also curious why representation theory is much harder for the nonzero characteristic, since this is an area I am currently focused on. – nobody Apr 23 '17 at 23:02
• In group representation theory, a fundamental theorem says that every representation of a group $G$ over a field $F$ can be decomposed into a direct sum of irreducible subrepresentations, and its proof requires the ability to divide by the order of $G$ within the field $F$, which isn't always possible if $F$ has positive characteristic. – JDZ Apr 23 '17 at 23:58