What are subfields of $\mathbb{C}$?

Question

I only took the first undergraduate abstract algebra course, so i don't know (at all) what Galois theory is about.

I'm asking this question since i'm not sure of the definition of inner product space and normed space.

Generally, they are defined on a vector space over $\mathbb{R}$ or $\mathbb{C}$, but some contexts define them on a vector space over a subfield of $\mathbb{C}$.

Are $\mathbb{Q},\mathbb{R},\mathbb{C}$ all the subfields of $\mathbb{C}$?

Answer 1

There are lots of subfields of $\mathbb{C}$. Some typical examples:

• The algebraic closure $\overline{\mathbb{Q}}$ of $\mathbb{Q}$.
• $\mathbb{Q}(\sqrt[n]{a})$ for some $n \in \mathbb{N}$ and $a \in \mathbb{Q}$. In particular $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(i)$.
• The cyclotomic fields $\mathbb{Q}(\zeta_n)$, where $\zeta_n$ is a primitive $n$th root of unity.
• The field of constructible numbers.
• The examples mentioned so far are algebraic over $\mathbb{Q}$. A non-algebraic example is $\mathbb{Q}(\pi)$.

Using the axiom of choice, one can even construct "arbitrary complicated" subfields of $\mathbb{C}$. Generally speaking, many fields may be embedded into $\mathbb{C}$. They only have to be of characteristic zero and should have cardinality $\leq |\mathbb{C}|$.

Answer 2

No; for example, $\Bbb Q (\sqrt{2}) = \{a + b\sqrt{2}\;|\;a,b \in \Bbb Q\}$ is a subfield of $\mathbb{C}$. Any field extension of $\Bbb Q$ that is a subset of $\Bbb R$ is necessarily a subfield of $\mathbb{C}$.