Show that $x^2 +1$ is irreducible in $\mathbb{R}[x]$, but it has roots in $\mathbb{R}[x]\space/\space(x^2 +1) \cong \mathbb{C}$

Question

So I know that for something to be irreducible, then it cannot be written as the product of non-constant polynomials of smaller degree, but I don't know how to show that the factors don't exist is the reals. From my traditional understanding of mathematics, obviously it makes sense, but I don't know how to show it abstractly, and I don't know how to show that they exist in the set of complex numbers.

Answer 1

If $r$ is a root of a polynomial $p(x)$ then $x-r$ is a factor of that polynomial. Conversely, if a polynomial has a linear factor then it has a root.

With that information you should be able to prove that $x^2 + 1$ is irreducible over the reals.

If you think a little bit you should be able to find a root of that polynomial in $\mathbb{R}[x]\space/\space(x^2 +1)$. (It is almost staring you in the face.)

Answer 2

Take the element $x + \langle 1 + x^2 \rangle \in \Bbb{R}[x] / (x^2 + 1)$. Let $p(x) = x^2 + 1$. Then \begin{align*} p(x + \langle 1 + x^2 \rangle) &= (x + \langle 1 + x^2 \rangle) \cdot (x + \langle 1 + x^2 \rangle) + (1 + \langle 1 + x^2 \rangle) \\ &= x \cdot x + (x + \langle 1 + x^2 \rangle) + 1 + \langle 1 + x^2 \rangle \\ &= (x^2 + 1) + \langle 1 + x^2 \rangle \\ &= 0 + \langle 1 + x^2 \rangle, \end{align*} which is to say, $x + \langle 1 + x^2 \rangle$ is a root of $p$.

Answer 3

Well, let $I=\langle x^2+1\rangle$. The quotient ring ${\Bbb R}[x]/I$ contains a zero $\bar x =x+I$ of $x^2+1$, since $\bar x^2 + 1 = (x+I)^2+1 = (x^2+1)+I = \bar 0$ in the quotient ring.

Thus the quotient ring consists of the elements $a+b\bar x$, $a,b\in{\Bbb R}$. Addition is componentwise $$(a+b\bar x) + (c+d\bar x) = (a+c) + (b+d)\bar x$$ and multiplication requires that $\bar x^2+1=0$ in the quotient ring, $$(a+b\bar x) (c+d\bar x) = (ac-bd) + (ad +bc)\bar x.$$

The mapping ${\Bbb R}[x]/I\rightarrow {\Bbb C}: a+b\bar x \mapsto a+bi$ is a ring isomorphism, where $i$ is the imaginary unit with $i^2+1=0$.