Show that every algebric extension over $\mathbb{R}$ has degree $1\ \text{or}\ 2$

Question

Show that every algebric extension over $\mathbb{R}$ has degree $1\ \text{or}\ 2$

i know that algebric closure of $\mathbb{R}$ is $\mathbb{C}$. which have dim $2$ over $\mathbb{R}$.then can i use this to show algebric extension over $\mathbb{R}$ has degree $1\ \text{or}\ 2$ ??

Answer 1

Since $\mathbb C$ is algebraically closed, it has no proper algebraic extensions. (Can you see why?)

Therefore, the only algebraic extensions of $\mathbb R$ are intermediate extensions $\mathbb R \subset K \subset \mathbb C$. The degree of the extension $K/\mathbb R$ must divide $2$, and is therefore $1$ or $2$.

Answer 2

Hint

The only polynomials that are irreducible over $\mathbb{R}$ are of degree $1$ or degree $2$.