What are the prime ideals of the polynomial ring $\mathbb{R}[x]$?

Question

The exact question is:

And the solution is

What I don't get in this solution is the circled part. Why are there no other irreducible polynomials in $\mathbb{R}[x]$?

Isn't $x^4+1$ irreducible in $\mathbb{R}[x]$?

Answer 1

$$x^4+1=(x^2+\sqrt2 x+1)(x^2-\sqrt2 x+1)$$

More generally, if $f$ is a real polynomial with a non-real complex zero $\alpha$, then $f$ has the irreducible real factor $(x-\alpha)(x-\overline\alpha)$.

Answer 2

First of all, any polynomial of odd degree in $\Bbb R[x]$ is reducible because it has a real root, by the intermediate value theorem.

Any polynomial of even degree greater than $2$ cannot be irreducible since $\Bbb C = \Bbb R(i)$ is an algebraic closure of $\Bbb R$ and an extension of degree $2$.