Is Complex Numbers the biggest field? If yes, is there any easy proof to understand it?

Question

1. Is the Complex Numbers the biggest field?
2. If yes, does anyone have a "simple"/"easy to understand" proof?

Answer 1

It depends on your notion of "biggest". The complex numbers aren't the biggest field in the sense that you could adjoin a new variable to them, say $\mathbb{C}(x)$, and obtain a new field into which $\mathbb{C}$ sits embedded as a subfield. They also aren't the biggest field in terms of cardinality, since you could probably put field structures on sets of arbitrarily large cardinalities.

But they are biggest, in the sense they are algebraically closed. This means that if you try to adjoin a new element to them that satisfies some polynomial relation, you... can't. The variable $x$ in $\mathbb{C}[x]$ doesn't satisfy any relations, so this extension isn't algebraic. But something like $\mathbb{C}[x]/{(3x^2 + 1)}$, where we add in a "new" thing $\alpha$ satisfying the polynomial equation $3\alpha^2 + 1 = 0$, is an algebraic extension. But you know that every complex polynomial has a root, so we can't add in this $\alpha$ to $\mathbb{C}$ - it's already there. On the other hand, $\mathbb{R}[x]/{(3x^2 + 1)}$ is "bigger" than $\mathbb{R}$ since we added in a new thing - a solution to $3\alpha^2 + 1 = 0$. This didn't already live in $\mathbb{R}$, so adding this thing actually extends the field.

Answer 2

It depends on the interpretation of the word 'biggest'. There are certainly fields of larger cardinality than $\mathbb C$ that contain $\mathbb C$ (for instance, take any set $S$ and consider the field $\mathbb C(S)$ where the elements of $S$ are adjoined as independent transcendental elements). So, if you measure size by cardinality, then for every field $\mathbb F$ there is a larger field containing it, and thus, in particular, $\mathbb C$ is not largest.

However, if by biggest you mean that every polynomial over it has a root in it, in other words being algebraically closed, then $\mathbb C$ is well-known to be algebraic closed and thus in that sense $\mathbb C$ is the largest algebraic extension of $\mathbb R$.