Is this an example for $[E:\Bbb R]$=4?

Question

Consider the polynomial $x^4+1$. Note that this is irreducible in $\Bbb R[x]$. Therefore the quotient $\Bbb R[x]/ <x^4+1>$ is a field, which is a degree 4 extension over $\Bbb R$. However, from this question (No extension to complex numbers?), and from intution, it seems that this is impossible because $[\Bbb C:\Bbb R]=2$. This is sort of out of my head. Can someone help me clear this up? Thanks.

Answer 1

Note that this is irreducible in $\Bbb R[x]$

$$x^4+1=(x^2-\sqrt2x+1)(x^2+\sqrt2x+1)$$

Answer 2

Who says it's irreducible?

$$(x^2 - \sqrt{2} x + 1)(x^2 + \sqrt{2} x + 1)$$