Explicit example of an algebraically closed field of cardinality greater than continuum.

Question

Is there an explicit example, in ZFC, of an algebraically closed field whose cardinality is greater than the cardinality of the set of real numbers? I would be very interested in such an example.

Answer 1

Here is a construction: For any set $X$, consider the ring $\mathbb{Z}[X] = \mathbb{Z}[(x)_{x\in X}]$, the ring of polynomials with integer coefficients and variables from $X$. Alternatively, this is the free ring generated by $X$. It is a domain, and its fraction field is $\mathbb{Q}(X) = \mathrm{Frac}(\mathbb{Z}[X])$, the field of rational functions with variables from $X$. Let $\overline{\mathbb{Q}(X)}$ be the algebraic closure of $\mathbb{Q}(X)$.

Then $\overline{\mathbb{Q}(X)}$ is an algebraically closed field of characteristic $0$ and cardinality $\max(|X|,\aleph_0)$. In particular, if $|X|>2^{\aleph_0}$, then $|\overline{\mathbb{Q}(X)}| = |X|>2^{\aleph_0}$.

In fact, every algebraically closed field $K$ of characteristic $0$ is isomorphic to one of the form $\overline{\mathbb{Q}(X)}$, where $X$ is a transcendence basis of $K$ over $\mathbb{Q}$.

Similarly, every algebraically closed field $K$ of characteristic $p$ is isomorphic to one of the form $\overline{\mathbb{F}_p(X)}$, where $X$ is a transcendence basis of $K$ over $\mathbb{F}_p$.