Example of infinite extension of $\mathbb{R}$?

Question

Isn't any extension of $\mathbb{R}$ either $\mathbb{R}$ or $\mathbb{C}$ ?

Answer 1

Yes, of course there are plenty of infinite extensions of $\mathbb{R}$ or $\mathbb{C}$ -- as large as you like.

The most commonly studied example would be $\mathbb{R}(t)$, where $t$ is a new variable. This is the field of rational functions (a polynomial over a polynomial) in $t$. This is an infinite extension of $\mathbb{R}$.

We can make ever larger extensions of $\mathbb{R}$ by taking $\mathbb{R}(t_1, t_2, t_3, \ldots t_k)$, or even $\mathbb{R}(T)$ where $T$ is some countable, or uncountable, or larger set of new variables. This will be the set of expressions $\frac{p}{q}$, where $p$ and $q$ are polynomials in those variables.

Another example is the Hyperreal numbers. This is an uncountable field containing the real numbers that is nevertheless much larger, including infinitesimal and infinite numbers.

In general, your intuition should be to assume that arbitrarily large objects of a certain kind will exist, rather than to assume otherwise. This is the case with almost every object you have probably heard of: groups, rings, fields, totally ordered sets, vector spaces, topological spaces, metric spaces, and so on. Specifically, if there is an infinite object of that kind, and the definition of such objects is simple enough (first-order, but don't worry about that) then there will be examples of that object of arbitrary large size. This is a basic result in model theory.

Answer 2

The answer of @6005 is great, but I also wanna add the quaternion field $\mathbb H$, which could be consider as an extension of $\mathbb R$:

$$\mathbb H=\mathbb R[i,j,k]$$

where

$$i^2=j^2=k^2=-1$$

and such other rules of this type.

It is just one example among many others.