In his doctoral thesis, in 1799, Gauss proved that every polynomial of degree $n$ has $n$ roots, as long as complex numbers are accepted. For example, $x^2 + 1 = 0$ has no real solution, but has two complex ones:

$$x^2 + 1 = (x - i)(x + i).$$

Gauss also popularized the way of drawing complex numbers as points on a plane, the plane complex: $a + bi$ is the point $(a, b)$. Norwegian Caspar Wessel (1797) and Swiss Jean-Robert Argand (1806) had the same idea, but almost no one read their work; was the fame of Gauss, from 1831, who spread the drawing. That’s why it is also called Argand-Gauss plane. And then a nice connection appears with the polygon with 17 sides. The three roots of $x^3 = 1$ are

$$1, \qquad -\frac{1}{2} + \frac{\sqrt{3}}{2}\,i, \qquad -\frac{1}{2} - \frac{\sqrt{3}}{2}\,i,$$

and, in the plane, they are the vertices of a equilateral triangle inscribed in the circle of radius 1. Of the same way, the 17 roots of $x^{17} = 1$ are the vertices of the 17 sided polygon. I studied this equation that Gauss arrived at the construction.

All topics by Carl Friedrich Gauss · ← Clock arithmetic · Counting prime numbers →

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