Are there any number fields that extend $\mathbb C$?

Question

If there are number fields that extend $\mathbb C$, what are some examples?

Answer 1

A number field is, by definition, a finite extension of the rationals and so the simple answer to your question is "no".

Moreover, the field of complex numbers is itself closed. In other words all polynomials over this field completely factorise and so the complex numbers cannot be extended in the way you are probably thinking.

There are however larger algebraic structures which do contain the complex numbers. For example, the complex number $a+bi$ can be thought of as the 2x2 matrix $$\begin{pmatrix}a&-b \\b&a\\\end{pmatrix}$$ and these matrices are contained in the much larger set of all 2x2 matrices with real entries.

As pointed out by @JMoravitz the complete set of such 2x2 matrices is not a field since their multiplication is not (in general) commutative.

Answer 2

Depends on what you mean as a number, but both the field of rational functions, holomorphic functions, or linear operators on them, are extensions of complex numbers, because they include complex numbers as constants.