What is the essential difference between real number and complex number?

So I'm learning linear algebra and we started by learning fields, vector space and then we just started linear map.

An interesting thing I noticed is that lots of nice propositions about vector space are true over real number field.

Example 1: for three subspaces of a vector space, their union is a subspace if and only if one of the subspace contains the other two.

Generally this is not true, over the field of 4 elements we can construct a counter example. However if the vector space is over real number, then it's true. Hopefully I didn't make any mistake.

Example 2: the conjugation of complex number from C to C is a linear map over real number but not linear over complex number. Furthermore, if a map between two vector spaces over real number is additively linear, then it is also scalar multiplication linear. I.e. A linear transformation.

So I feel like there's something interesting going on behind real numbers and complex numbers. Even though they are both field, they're somehow different? What branch of math studies the nature of such things, like nature of fields?

A difference between the real numbers and the complex numbers is the fact that the complex numbers are algebraically closed. Being algebraically closed means that any polynomial has a root, or equivalently that every polynomial factors into linear terms, for example $$x^2+1 = (x+i)(x-i)$$ $$x^2 - i = (x + {1 \over \sqrt{2}} (1+i))(x - {1 \over \sqrt{2}} (1+i))$$

The complex numbers being algebraically closed is known as the fundamental theorem of algebra. This has a few consequences in linear algebra. For example, because the characteristic polynomial of an $n \times n$ matrix has exactly $n$ linear factors up to multiplicity, there are exactly $n$ eigenvalue up to multiplicity. It follows that every matrix over the complex numbers is diagonalizable (because there are $n$ eigenvalues to work with), which is not true over the real numbers.

Another way fields can differ is the notion of finiteness. The field with 4 elements, has, well, 4 elements, and so one can list out all $n \times n$ matricies over it, and pick out finite subspaces that union to the whole thing.

These examples belong to the realm of algebra. They point out how differences in fields can cause differences in the structure of vector spaces over the field. In fact, one can generalize to matricies over an arbitrary ring, and try to develop as much linear algebra as possible. The study of linear algebra over a ring can lead to results such as the classification of finitely generated abelian groups.

Algebra studies the nature of objects like fields, and the starkest difference between the real and complex numbers is in their symmetries (or lack thereof). The only automorphism of the reals is the trivial automorphism. The complex numbers have as many automorphisms as there are subsets of the real line!

• Maybe not the most significant difference between the two, but surely among the most spectacular. Commented Apr 18, 2018 at 2:44
• So, abstract algebra is what I need to learn? I'm really not that talented in pure abstract math, which is why I choose to major applied math. But my curiosity is killing me. I want to figure out everything I don't know and trace to the very basic of logic and universe badly. Sadness of a stupid person :( Commented Apr 18, 2018 at 2:47
• @FluffySkye: Yup, and I can tell you that curiosity will finally be leading you to want to learn all fields of mathematics. You should be happy. Cheers~ Commented Apr 18, 2018 at 4:46
• Is f(x) = -x not a nontrivial automorphism of the reals? Commented Apr 18, 2018 at 5:04
• It is not. Field automorphisms need to take 1 to 1. Commented Apr 18, 2018 at 10:49