How to list down all the prime ideals of R[x](R=real number)?
I know that R[x] is a PID,thus the ideal generated by every irreducible polynomial in R[x] is a prime ideal.
But can we write it in set notation,just like we can say for C[x](C=complex number)?
Let $f(X)$ be an irreducible polynomial in $\mathbb R[X]$. We shall show $\deg f(X)\leq 2$. Suppose $f$ has no real root or in other words $f$ is not linear. By the Fundamental Theorem of Algebra, $f$ has a root $\alpha \in \mathbb C$. Since $\alpha \notin \mathbb R$, $\bar \alpha$ is also a root of $f$. This shows $(X-\alpha)(X-\bar \alpha)$ divides $f(X)$ in $\mathbb C[X]$. But $(X-\alpha)(X-\bar \alpha )$ is a real polynomial. So we get $(X-\alpha)(X-\bar \alpha)$ divides $f(X)$ in $\mathbb R[X]$. Thus $$f=\mu(X-\alpha)(X-\bar \alpha)$$ where $\mu$ is a unit in $\mathbb R[X]$ i.e. $\mu \in \mathbb R^*$.
Thus the irreducible polynomials are precisely linear polynomials and quadratics with no real roots.