Is there a field extension over the real numbers that is not the same as the field of complex numbers?

Solution 1:

There is not. The extension $[\mathbb C : \mathbb R]$ has degree 2, so there cannot be a proper intermediate extension.

Assume such an extension $F$ exists. Then by the tower theorem, we have $[\mathbb C:F][F:\mathbb R]=2$. For it to be a proper extension, each factor must be greater than one. This is a contradiction.

Solution 2:

Suppose $F$ is a field such that $\mathbb{R} \subsetneq F \subseteq \mathbb{C}.$ Then $F$ contains at least one element $a+bi$ where $b\neq 0.$ Since $F$ contains all the reals, it contains $b^{-1}$ and $-a,$ so it contains $b^{-1}\left((a+bi)-a\right)=i.$ Hence it contains all complex numbers and $F=\mathbb{C}.$