What are the irreducible elements in $\mathbb{R}[x]$ and $\mathbb{C}[x]$?

Question

What are the irreducible elements in $\mathbb{R}[x]$ and $\mathbb{C}[x]$?

For $\mathbb{C}[x]$, I guess the irreducible elements are all polynomials $x-z$ where $z\in \mathbb{C}$.

For $\mathbb{R}[x]$, I guess the irreducible elements are all polynomials $x-a$ and some other polynomials?

Answer 1

For $\Bbb C[X]$, you also have the polynomials $aX+b$ ($a\neq 0$) but no other since $\Bbb C$ is algebraically closed.

For $\Bbb R[X]$, you also have the polynomials $aX+b$ ($a \neq 0$) and some degree 2 polynomials $aX^2+bX+c$ ($a \neq 0$), but that's all since $[\Bbb C:\Bbb R]=2$. Notice that for a polynomial of degree $\leq 3$, irreducibility is equivalent to "having no roots", so $aX^2+bX+c \in \Bbb R[X]$ is irreducible iff $b^2-4ac<0$.

Answer 2

For $\mathbf C[X]$, you're (almost) right: the irreducible polynomials are linear polynomials $\;c_1X+c_0$, $c_0, c_1\in\mathbf C$. Such a polynomial is associated to a unique monic polynomial. This results from the D'Alembert-Gauß theorem.

As for $\mathbf R[X]$, as the complex roots of a polynomial with real coefficient are pairwise conjugate, there results irreducible polynomials are linear polynomials and quadratic polynomials with a negative discriminant.

Answer 3

Let be $R[x]$ is a polynomial ring. If $f(x)\in R[x]$ is an irreducible polynomial, then $(f(x))$ is a prime ideal.

we know that $\mathbb R[x]$ and $\mathbb C[x]$ are $PID$ then every prime ideal is maximal.

In $\mathbb C[x]$ we know that maximal ideal is the form $(x-a)$ , then irreducible polynomial in the $\mathbb C[x]$ are the form $f(x)=x-a$.

In $\mathbb R[x]$, $f(x)$ is irreducibe polynomial if $(f(x))$ is a maximal ideal