Applications: Constructions with a straightedge and compass

Figure 5.1: A geometric construction of a trisection

For the purposes of this article, assume that any references to a straightedge is that of an unmarked straightedge. A marked straightedge could be exploited to violate some of the properties that will be established for geometries constructed by a unmarked straightedge, so the marked straightedge is not considered in this article.

One of the interesting results of Galois groups is the proof that using an unmarked straightedge and compass, one can not trisect an angle. In order to understand how this proof works using Galois groups, one must also know what a constructible number is, along with understanding the limitations that a straightedge and compass pose.

A constructible number has a definition that may be familiar to those who are familiar with the straightedge and compass: it is a number in ℝ with the property that it can be found through the constructions made by a straightedge and compass. In the context of fields, this definition becomes: a number is a constructible number if it lies with the field extension of the rational numbers,

Q[a_i, ...]

So how does one make any sort of geometric statement about angles using this group notation? Well, the answer to this question involves exploiting properties of polynomials and their relationships with their Galois group. Consider an angle, 30 degrees. Now consider it within a cosine as a cos(30 degrees). The solution to the cosine of 30 degrees is simply:

cos(30 degrees) = √3/2

To convince oneself that this proof works for the angle just as well as the cosine of that angle, think that the angle could not be constructed if the cosine of that angle could not be constructed. cos(30 degrees) can be written in terms of cos(20 degrees) by applying the triple angle identity for cosine:

cos(3θ) = 4cos³(θ)−3cos(θ)

Plugging this in for cos(30 degrees) in a parameterized form,

4cos³(θ)−3cos(θ) = √3/2

By letting h = cos(θ),

4h³ -3h - √3/2 = 0

If one lets h = cos(10 degrees), it is easy to see that the polynomial will solve itself. However, note that in order to be a constructible number, the polynomial must have rational roots. The use of the Galois group using the splitting field of this function can prove that this polynomial does not have any rational roots, making it impossible to perform the trisection of 30 degrees with an unmarked straightedge and a compass, as the cosine of the trisected angle 10 degrees is not constructible. This may seem like an odd application for a very abstracted concept, but note that the Greeks had been trying to solve sweeping questions such as this roughly a thousand years before it was proven, but their mathematics was not as rigorous as it is in the modern day.

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