3.2 Galois Groups Review

summary

Back to course

Symmetry is inherent to number theory.

So far we have seen how symmetry plays out in geometry, and how symmetry groups are useful in quantum mechanics.
Now we want to take a look at how the same symmetry groups are useful in the study of numbers.
Number theory is yet another “continent” of mathematics that can be unified using symmetry.

What kinds of numbers exist?

There are whole numbers, or integers (…-3, -2, -1, 0, 1, 2, 3…) and rational numbers (fractions) $\frac{s}{r}$ where $s$ and $r$ are integers, such as 17/31 or 9/25.
For a long time, mathematicians thought these were the only two kinds of numbers. Pythagoras believed only rational numbers could exist, and any number that could not be expressed as a ratio of two integers was not a “legitimate” number somehow.
Of course we know now that there are irrational numbers that we can understand in physical reality. The number $\sqrt 2$, for example, is simply the diagonal of a square with length=1, and it becomes apparent that this number cannot be written as the ratio of two integers.
We can use irrational numbers to create “new” numbers such as $$a+b\sqrt 2$$where $a$ and $b$ are rational numbers. We can add and multiply irrational numbers the same way we do rational numbers.
These “new” numbers form novel numerical systems called number fields.

Number fields contain hidden symmetries.

In our example, there is the following symmetry:$$a+b\sqrt 2 \longleftrightarrow a-b\sqrt 2$$These are both solutions to the same equation of degree 2.
The group of symmetries of a number field is called a Galois group, in honor of Evariste Galois, a brilliant French mathematician who died in a duel at age 20.
Galois was interested in trying to solve polynomial equations with radicals. We have learned about the general algebraic solution for a quadratic equation $ax^2+bx+c=0$:$$x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$$
The important thing to note in this formula is that taking a square root is involved. This formula was published as early as the year 830. Mathematicians similarly devised general algebraic solutions for cubic equations (degree 3) and quartic equations (degree 4) in the 16^th century, using radicals of degrees 3 and 4, respectively.
But then for almost 300 years, nobody could devise a general algebraic solution for a polynomial equation of degree 5 (quintic) and higher.$$ax^5+bx^4+cx^3+\dotsb =0$$
This mystery was solved by Galois, who claimed that this solution couldn’t be discovered because it didn’t exist for such order polynomials. Galois explained a fundamental difference between polynomials of degree 2-4 and polynomials of degree 5 and higher, using what we now call Galois groups.
Instead of trying to solve the equation generally, Galois looked at the group of symmetries of its solutions. From its structure, we can discern whether there is a formula for solutions in terms of radicals or not.