$x^4 + 1$ reducible over $\mathbb{R}$... is this possible?

Question

I am seeing this on a homework and am wondering if this is a typo. I am aware that $x^4 + 1$ is irreducible over $\mathbb{Q}$. I know the following:

A polynomial being irreducible over some ring does not imply that it is irreducible over some superring.

$x^4 + 1$ has no linear factors

and if it has no linear factors, then it can only be factored into polynomials of degree 2, and for there to be no term $a_1x^1$, the factor must be of the form $a_2x^2 + a_0$, and this can be done in $\mathbb{C}$ but not in $\mathbb{R}$.

Answer 1

Remember Sophie Germain: $$ x^4+4 = (x^4+4x^2+4)-(4x^2) = (x^2+2)^2-(2x)^2 = (x^2+2x+2)(x^2-2x+2) $$ and adjust the above line by replacing $x$ with $\sqrt{2}\,x$.

Answer 2

Even if you don't know the explicit factorization, you certainly know that a quartic polynomial in $\mathbb{R}[X]$ is reducible.

Indeed, a consequence of the fact that every polynomial of positive degree in $\mathbb{C}[X]$ has a root, we can see that the irreducible polynomials over $\mathbb{R}[X]$ are those of degree $1$ and the quadratic polynomials with negative discriminant.

To wit, let $f(X)\in\mathbb{R}[X]$ be monic of degree $>1$. If $f$ has a real root, then it is reducible. So, assume it has no real root. Since it has a complex root $a$, also $\bar{a}$ is a root, because $$ f(\bar{a})=\overline{f(a)}=\bar{0}=0 $$ (overlining means conjugation). Thus $f$ is divisible (in $\mathbb{C}$) by $X-a$ and $X-\bar{a}$, so also by their product $$ g(X)=(X-a)(X-\bar{a})=X^2-(a+\bar{a})X+a\bar{a} $$ (because $X-a$ and $X-\bar{a}$ are coprime in $\mathbb{C}[X]$). However, $g(X)$ has real coefficients, so the quotient of the division of $f(X)$ by $g(X)$ is again a polynomial in $\mathbb{R}[X]$. So, unless $f$ has degree $2$ (and so $f(X)=g(X)$), $f$ is reducible.

Answer 3

The four complex roots of $X^4+1$ are the 4th roots of $-1=e^{\pi i}$, i.e. $$ z_1=e^{\frac14\pi i},\quad z_2=e^{\frac34\pi i},\quad z_3=e^{\frac54\pi i},\quad z_4=e^{\frac74\pi i}. $$ Note that $$ z_4=\bar z_1\quad\text{and}\quad z_3=\bar z_2. $$ Thus $P(X)=(X-z_1)(X-z_4)$ and $Q(X)=(X-z_2)(X-z_3)$ are polynomials with real coefficients and $$ X^4+1=P(X)Q(X). $$

Answer 4

To understand where you went wrong, note that '$x^4+1$ has no linear factors' doesn't imply 'there is no term linear in $x$ in a factor'. Indeed, the restrictions on terms in the factorization can be derived straightforwardly: suppose that $x^4+1$ is a product of two quadratics $a_2x^2+a_1x+a_0$ and $b_2x^2+b_1x+b_0$. It should be obvious that via multiplication by scalars we can force $a_2=b_2=1$; now, the product is $x^4+(a_1+b_1)x^3+(\text{some complicated term in }x^2)+(a_0b_1+b_0a_1)x+a_0b_0$. Comparing terms, we get $a_1=-b_1$ (from the $x^3$ term) and then $a_0=b_0$ (from the $x$ term, using the new knowledge about $a_1$ and $b_1$) and $a_0b_0=1$ (from the constant term). The last two obviously imply that either $a_0=b_0=1$ or $a_0=b_0=-1$, so the product is either $(x^2+a_1x+1)\cdot(x^2-a_1x+1)$ or $(x^2+a_1x-1)\cdot(x^2-a_1x-1)$. Now you can go back to the 'complicated term in $x^2$'; you should find a simple quadratic equation in $a_1$ that you can solve.

Answer 5

We know that $x^4+1=0$ has four solutions in $\mathbb C$, none of which are real. We also know that complex solutions to a polynomial equation with real coefficients come in conjugate pairs. So the solutions must be some set of the form $ \{w, \bar{w}, z, \bar{z} \}$, which means that it factors into complex linear factors as: $$x^4 + 1 = (x-w)(x-\bar w)(x-z)(x-\bar z)$$ Now if you multiply these together in pairs, you get two quadratic factors with real coefficients. So even without knowing what the factors are, we know that $x^4+1$ factors into two quadratics with real coefficients.

You can go even further with this if you know what the complex roots are, of course. Let $w=e^{\pi i/4}$, which is certainly one of them. The other three are then found by multiplying by $i$: $z = iw = e^{3 \pi i / 4}$, $\bar z = - w = e^{5 \pi i/r}$, and $\bar w = -iw = e^{- \pi i/4}$. Then you can work out explicitly what the real quadratic factors are.