The Universe of Discourse

All polynomials of degree 3 or greater factor over the reals

A couple of years ago I wrote:

One day when I was in high school, I bumped into the fact that !!\sqrt{7 + 4 \sqrt 3}!!, which looks just like a 4th-degree number, is actually a 2nd-degree number. It's numerically equal to !!2 + \sqrt 3!!. At the time, I was totally boggled.

I had a kind of similar surprise around the same time in connection with the polynomial !!x^4+1!!.

Everyone in high school algebra learns that !!x^2-1 = (x-1)(x+1)!! but that !!x^2+1!! does not similarly factor over the reals; in the jargon it is irreducible.

Every cubic polynomial does factor over the reals, though, because every cubic polynomial has a real root, and a polynomial with real root !!r!! has !!x-r!! as a factor; this is Descartes’ theorem. (It's easy to explain why all cubic polynomials have roots. Every cubic polynomial !!P(x)!! has the form !! ax^3!! plus some lower-order terms. As !!x!! goes to !!±∞!! the lower-order terms are insignificant and !!P(x)!! goes to !!a·±∞!!. Since the value of !!P(x)!! changes sign, !!P(x)!! must be zero at some point.)

For example, $$\begin{align} x^3+1 & = (x+1)(x^2- x+1) \\ x^3-1 & = (x-1)(x^2+ x+1) \\ \end{align}$$

So: polynomials with real roots always factor, cubics always have roots, so cubics factor. Also !!x^2+1!! has no real roots, and doesn't factor. And !!x^4+1!!, which looks pretty much the same as !!x^2+1!!, also has no real roots, and so behaves the same as !!x^2+1!! so doesn't factor…

Wrong! It has no real roots, true, but it still factors over the reals:

$$x^4+1 = (x^2 + \sqrt2· x + 1) (x^2 - \sqrt2· x + 1)$$

Neither of the two factors has a real root. I was kinda blown away by this, sometime back in the 1980s.

The fundamental theorem of algebra tells us that the only irreducible real polynomials have degree 1 or 2. Every polynomial of degree 3 or higher can be expressed as a product of polynomials of degrees 1 and 2. I knew this, but somehow didn't put the pieces together in my head.

Raymond Smullyan observes that almost everyone has logically inconsistent beliefs. His example is that while you individually believe a large number of separate claims, you probably also believe that at least one of those claims is false, so you don't believe their conjunction. This is an example of a completely different type: I simultaneously believed that every polynomial had roots over the complex numbers, and also that !!x^4+1!! was irreducible.

[ Addendum 20220221: I didn't remember when I wrote this that I had already written essentially the same article back in 2006. ]

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