What does "characteristic" mean in mathematics?

In Germany, I have heard "ABC is 'characteristic' for XYZ" sometimes from math students. It was used like "if you know ABC, then you know you're talking about XYZ" or "ABC describes XYZ completely".

For example:

The generator is characteristic for a cyclic group.

I know that the characteristic polynomial $CP_A(\lambda) := \det(\lambda E_n - A)$ does not completely describe the matrix $A$ as similar matrices have the same characteristic polynomial. So here "characteristic" seems to mean something like "an important attribute".

Now I've learned the term characteristic of a unit ring $(R, +, \cdot)$:

It exists exactly one ringhomomorphism $\varphi: \mathbb{Z} \rightarrow R$. Let $\text{char}(R) := n$ be the non-negative generator of the kernel.
According to Wikipedia, it is $$\text{char}(R) := \min\{n \in \mathbb{N} | \underbrace{1 + \dots + 1}_{n} = 0\} \text{ (or 0 if no such $n$ exists)}$$

I've also seen that the term characteristic function exists (but I don't know what it is).

Questions

So my questions are:

How do mathematicians use the word "characteristic"? What do the three uses I described have in common?
In how far is the characteristic of a unit ring interesting? Why is the characteristic of a unit ring "characteristic"? What can you say about the ring, when you only know the characteristic?

Solution 1:

So for the first question, one answer could be

They don't have to have anything in common, although a couple of them do. They don't all stem from a single source, and their uses depend on which field you are in.

Generally

In just plain English, the phrase "is characteristic of" means "is a distinguishing feature of." By extension in mathematics, it could be used this way to describe something that completely describes another thing. The generator of a cyclic group is a good example of this, since you can recover the entire group from a single generator.

More generally, mathematicians like to talk about "what characterizes" a certain thing. So, for example, the Artin-Wedderburn theorem is a characterization of the semisimple rings as finite products of matrix rings over division rings. The Ore condition is what characterizes which domains can be densely embedded in division rings.

Special uses

Characteristic subgroups in group theory are very special normal groups. Essentially, they are unmoved by automorphisms of the group. This makes them even more special than normal subgroups.

The characteristic polynomial doesn't describe the matrix, but it does characterize information about the transformation that the matrix represents. Really the particular matrix representation is secondary to the transformation.

Sometimes eigenvectors and eigenvalues are referred to as "characteristic" rather than "eigen." (This also suggests somewhat that "characteristic" is one of those overloaded math terms like "regular" or "normal" that pops up every time someone invents something new.)

The characteristic of a ring is important, but I don't know if I can tell you a single reason why. First of all, there are a lot of theorems that are first proven for characteristic zero. (I'm thinking in particular of some results in group rings.) This is usually because the very basic rings ($\Bbb Z,\Bbb R,\Bbb Q,\Bbb C$) are all characteristic zero, and maybe our intuition is better with them. Ordered fields, which are in some way bound up with our intuition of geometric length, are all characteristic zero too.

When the characteristic is nonzero, things are harder because you have to cope with a kind of (very interesting!) degeneracy. The characteristic 2 case seems like the "least nice" because of its appearance in some geometrically freak cases. Even though I've said "degeneracy" and "freak" now to describe these things, I still want to stress that they are interesting and important. Positive characteristic fields are the natural environment for algebraic coding theory, after all :)

The first time I encountered characteristic functions was in measure theory, where the characteristic function of a set is one which has value $1$ for elements of the set, and value $0$ elsewhere. Perhaps the motivation for calling it "characteristic" is that it clearly distinguishes which points are inside the set and which points are outside the set.

Another use of "character" is the one from representation theory, where you talk about the character afforded by a representation. I'm not aware of the true origins of the term, but I've always thought of it like this. The character of a representation is a distillation of some of the information carried by the representation. The information is distilled into a function from the group into a field. There, you can divine several important characteristics of the group. You might have also derived them directly from the representation, but the character might make things easier.

There is also something I know nothing about called the Euler characteristic which is a topological invariant. "Invariant" could be considered semantically close to "characteristic." They are both often used to describe qualities that are intrinsic to the object.

Solution 2:

These terms have no real mathematical link between them - i.e. they are all derived from the English meaning, but independently.

The first usage is much more common in the format "ABC characterizes XYZ". This means that XYZ is uniquely identified by the properties/characteristics ABC.

In the case of rings, the "characteristic" is a particular characteristic (in the English sense) of a ring - it's a property, but it's not a uniquely identifying one. This characteristic is quite important - for example in the case of vector spaces over fields, the characteristic of the field determines much of the behaviour of addition. When the characteristic is two, vector spaces behave somewhat strangely, stemming from the fact that $c = -c$ for scalars $c$. For example the skew-symmetric matrices ($M^T + M = 0$) do not necessarily have zeroes on the diagonal! When linear algebra is treated over arbitrary fields you thus often see "over a field with characteristic not two".