GALOIS THEORY: LECTURE 1
LEO GOLDMAKHER
After going over the syllabus and introducing ourselves, we began the course proper by describing six questions
I hope we will resolve by the end of this course.
(Q1) How does one trisect a given angle? How does one double a given cube?
Discussion.
Ancient Greek mathematicians were famously interested in geometry, especially straightedge
and compass constructions. Recall how these two tools work: given any two points P and Q, a
straightedge allows you to draw the line going through both P and Q, while a compass allows
you to construct the circle which passes through P and is centered at Q. Note that these are
instruments of construction, as opposed to instruments of measurement: a straightedge cannot
measure length (it’s not a ruler), and a compass cannot measure angle (it’s not a protractor).
It turns out that one can do a lot even with just these two primitive tools. For example, one
can draw a segment perpendicular to a given one; bisect a given line segment; bisect a given
angle; and double a square (i.e. given a square, construct a square with double the area).
There were three constructions the Greeks were notoriously frustrated by:
• Trisect a given angle
• Double a given cube
• Square a circle (i.e. construct a square whose area is the same as a given circle’s).
All three of these were open problems for about 2000 years.
Resolution.
In 1837, Pierre Wantzel proved the unexpected result that doubling the cube and trisecting the
angle were impossible (using only a straightedge and compass). The proof turns out to be an
easy consequence of the work of Galois. [The third problem – squaring the circle – also turns
out to be impossible. This follows from Lindemann-Weierstrass’ proof that π is transcendental.]
(Q2) How do you construct a regular n-gon?
Discussion.
Ancient Greek mathematicians could construct the first few regular n-gons using just compass and straightedge (with n = 3, 4, 5, 6), but didn’t know how to handle the case n = 7.
They also knew how to turn a construction of a regular n-gon into a construction of a regular
2n-gon. By the end of the 18th century, mathematicians could construct regular n-gons for
n = 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, but the cases of n = 7, 9, 11, 13, 14 remained elusive.
Resolution.
In 1801 Gauss proved that if n is a product of distinct Fermat primes1, then the regular n-gon
is constructible. By the observation of the Greeks about the 2n-gon, this immediately implies
Date: September 10, 2015.
n
Let Fn := 22 + 1. A Fermat prime is any Fn which is prime. Unfortunately, Fermat primes are mysterious objects; all that’s
currently known is that Fn is prime for 0 ≤ n ≤ 4 and composite for 5 ≤ n ≤ 32.
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that any power of 2 times a product of distinct Fermat primes is also constructible. Using Galois theory, Pierre Wantzel was able to prove the converse statement: that if n is not a power
of 2 times a product of distinct Fermat primes, then the regular n-gon is not constructible.
(Q3) What is the quintic formula?
Discussion.
Recall that any quadratic equation can be solved using the quadratic formula. The aspect of
this formula I’d like to highlight is that it involves precisely five operations: +, −, ×, ÷, and
√
. A cubic formula exists as well. Although substantially messier than the quadratic formula, it involves only one additional operation: the cubic formula can be expressed purely in
√
√
terms of +, −, ×, ÷, , and 3 . Similarly, the quartic formula (which is even messier than
√ √
√
the cubic formula) can be written in terms of +, −, ×, ÷, , 3 , and 4 . It’s now clear what
to expect about the quintic formula!
Resolution.
In 1799, Paolo Ruffini claimed that no expression involving only +, −, ×, ÷, and arbitrary
radicals will solve every quintic equation. However, his proof was 500 pages long (and, as
it turned out later, had gaps). In 1824, the 22-year-old Norwegian genius Niels Henrik Abel
published a six-page proof of Ruffini’s assertion; an expanded version appeared two years
later in Crelle’s journal.2
Neither Abel’s nor Ruffini’s proofs used Galois theory; indeed, they both predated Galois’
entry into mathematics3. However, a few years after Abel’s work, Galois would write down a
theory which gave much more precise results than Abel-Ruffini: given a polynomial, Galois
theory can be used to tell you whether or not the solutions to that particular polynomial can
be expressed in terms of standard arithmetic and radicals. Thus, for example, Galois theory
can prove that none of the solutions to the equation x5 − x − 1 = 0 can be expressed in terms
of arithmetic operations and radicals. By contrast, the Abel-Ruffini theorem doesn’t say anything about the solutions to this polynomial – only that there is no formula which solves every
quintic equation.
We will conclude our list of questions next time, and start in on describing Galois theory.
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Apparently Gauss ignored an early manuscript Abel sent him, dismissing it as the work of a crank.
Galois was born in 1811, so he was quite young when Abel’s proof appeared.
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