Why do we study the number of homomorphisms/isomorphisms between fields?

Question

From the first abstract algebra class, we encounter many problems that ask us to find the number of homomorphisms/isomorphism a between two algebra structures (e.g. field).

My first question is why do we study homomorphisms? To me, it is much weaker than isomorphisms (although stronger than a mere bijection, of course, so I guess it's better than nothing).

My second question is, why are we especially interested in the number of homomorphisms/isomorphisms? Or is it just something that tests students' understanding of the subject?

Update: Was out for some gyoza and wrote on my phone. I have this question because I just graduated and am taking an online course on Galois Theory. The quiz has quite some problems on "counting morphisms". One example would be,

Given an algebraic extension $F/\mathbb{Q}$, how many homomorphisms $F\to \mathbb{C}$ of fields are there?

Another example would be,

How many homomorphisms are there from $\mathbb{F}_{p^3}$ to $\mathbb{F}_{p^4}$?

I guess I am just not fully understanding the motivation/take away here (which really makes me think it's difficult to learn advanced math without being in a university). The first question sort of makes sense since, $\mathbb{C}$ is a "natural extension" of $\mathbb{Q}$ even though it is not an algebraic one. But I don't have any thought on the second question.

Answer 1

In a basic sense, algebra is about sets that have certain kinds of structures and about the functions that preserve those structures—namely, morphisms.

Morphisms reveal the structure of spaces. For example, if you know about the properties of one space $X$, the morphisms into another space $Y$ may tell you about $Y$.

As for why isomorphisms might not be as relevant: one way to look at it is that isomorphisms preserve too much structure to reveal interesting information about a space's structure —because if there is an isomorphism between two spaces, they are actually identical as far as their algebraic structure is concerned.

I am not sure why the number would be an especially important property to know about spaces, except that no matter what space you're looking at, you can always ask how many morphisms there are. The number of isomorphisms is interesting because it shows something about the symmetry of a space— how many ways one space can be mapped onto another.

Answer 2

As there is no one and only answer to your question, I will refer to the Galois Theory. If we have an extension $\mathbb{K}$ of a field $\mathbb{F}$ then the Galois group of $\mathbb{K}$ is the set of all the isomorphisms $\phi :\mathbb{K}\to \mathbb{K}$, that fixes $\mathbb{F}$ pointwise, in other words $\phi (c)=c \ \ \ \forall c\in\mathbb{F}$.

Answer 3

There are so much to say to fully answer to this question. I'll try to same some things but it is very likely I'll lots of stuff out.

Why do we study homomorphisms?

There are so many reason to studying homomorphisms between structures. Studying homomorphisms between structures we can find out a lot of properties of the structures themselves.

In what follows I'll try to give some unordered list of examples.

A typical way of studying a group structure is by looking at his homomorphisms inside the symmetric groups (group of bijections). By using the properties of these homomorphisms one can easily prove very often fact about the existence of subgroups, number of subgroups, order of elements etc etc etc.

In general if you have an algebraic structure $S$ by studying the surjective homomorphisms of the form $f \colon S' \to S$ you can prove equations that holds in $S$: that is because surjective homomorphisms preserves equations.

In linear algebra, linear applications, which are homomorphisms for vector spaces, allow us to study solution to linear equation with the language of vector spaces.

We know that any vector space has linear isomorphisms to has vector space of the form $\mathbb K^n$ (where $\mathbb K$ is a field and $n$ is a cardinal number). Through these isomorphisms we can reduce problems between vector spaces to problems in the algebra of matrixes.

We could go on on this, let's pass to the other question.

Why are we especially interested in the number of homomorphisms/isomorphisms?

Isomorphisms of a give structure form a group, the number of these isomorphisms is a first information that allows us to know better this groups.

Knowing group of isomorphisms for a given structure is really important because it gives information about the symmetry of the structure. From this information can get some invariants that can be helpful in proving that your structure is not isomorphic to some other structure.

In particular you can study these group by using group theory, and there are a lot of theorem that can give you informations on a group by simply knowing its cardinality.

I hope this helps.