Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
• The study of permutation groups is related to the roots of algebraic equations.
The study of groups is part of the larger discipline called abstract algebra. We have seen how group structure represents an abstraction of two seemingly different situations: an equilateral triangle and a stack of three pancakes. When we think of algebra, we normally think of typical algebraic problems such as"solve 3x +5 = 7". We use these types of algebraic statements to make general observations about how our number system works. To do this, we use variables to represent unknown numbers, thus freeing us from the constraints of specific numerical values so that we can see commonalities in different types of mathematical expressions and equations. For instance, we know that equations of the form y = mx + b all have something in common:, they give us straight lines that are completely characterized by their slope and y-intercept.
Certain equations have symmetry in the form of invariance under permutation. In other words, we can tell something about an equation in two variables by seeing what happens when we swap the variables. For example, the equation y = -x + 5 has the following graph:
If we permute x and y (i.e., swap their positions), we get x = - y + 5, which has the following graph:
The two graphs are the same! This is because the equation y = -x + 5 is invariant under permutation of x and y. By contrast, the equation y = x^{2} + 1 has the following graph:
The graph of its permutation, x = y^{2} + 1, looks like this:
This altered look shows that y = x^{2} + 1 is not invariant under permutation of x and y. This notion of permuting parts of equations will play a role as we attempt to answer a question similar to "what makes an equation solvable?"
If we combine the notions of abstraction seen in groups with the arithmetical power of algebra, we get a system of thought that brings the explanatory power of mathematics to things that are not numbers, such as the motions of symmetries and permutations. In short, abstract algebra is the study of how collections of objects behave under various operations. Its power lies in its generality.
Surprisingly, things such as symmetries and permutations can be used to understand problems from the realm of what we normally consider algebra. One of the first people to realize the vital link between symmetry and algebra was a 19^{th} century French mathematician named Evariste Galois.
Galois is a legendary figure, known for both his extraordinary insight and his dramatic life. A revolutionary, Galois was as much obsessed with politics as with mathematics, to the point that he was expelled from school. He spent time in prison, led protests, and all the while continued to do groundbreaking mathematics. Shot fatally in a duel at the age of 20, the young mathematician is said to have written down all of his mathematical knowledge in a letter the night before his demise. Whether or not this tale is true, the creative insight that Galois brought to mathematics is difficult to overstate.
Galois made an astonishing breakthrough while attempting to resolve one of the great questions of his age. Mathematicians had for centuries been trying to find general formulas that could give the roots to any polynomial using only the coefficients in the polynomial. Recall that a polynomial is a simple function that is a linear combination of powers of the input. For example, a quadratic equation is a second-degree polynomial such as p(x) = 3x^{2} + 7x - 2. The "roots" of p(x) are precisely the variable inputs that result in p(x) being 0. In this case, the famous quadratic formula tells you what the roots are in terms of the coefficients. Given a general quadratic equation:
p(x) = ax^{2} + bx + c
. . . the roots are:
and
Similar formulas exist for third-degree polynomials (cubics) and fourth-degree polynomials (quartics). For some time it had been a question of whether or not a fifth degree polynomial, or "quintic," was solvable, in the sense of having a similarly simple expression for the roots in terms of coefficients, square roots, cube roots, etc. and simple arithmetic. After years of work, mathematicians were able to find solutions only for specific cases, the simplest of these being quintic equations of the form ax^{5} + b = 0, for which the solution would be the fifth root of . Solutions that hold only for specific cases, however, are a far cry from the complete mastery that a general solution implies.
In 1824, the Norwegian algebraist Niels Henrik Abel published his "impossibility" theorem in which he proved that there is no general solution by radicals—no nice formula, in other words—for polynomials of degree five or higher. Earlier, in 1799, the Italian mathematician Paolo Ruffini had published a similar finding, though his proof was somewhat flawed. The fact that no general solution exists for quintic or higher-degree polynomials is now known as the "Abel-Ruffini Theorem" or, alternatively, "Abel's Impossibility Theorem." The methods that Galois used to reach a similar conclusion were more general, opening new doors to mathematical exploration. Where Abel and Ruffini showed simply that quintic and higher-degree polynomials could have no general solution by radicals, Galois showed why. Furthermore, his contribution was general enough to explain why polynomials of degree four and lower have general solutions and why those solutions take the form that they do. His ideas form the basis for what we now call Galois theory, the basis of modern group theory.
Galois' epiphany was to consider the symmetries of the roots of a polynomial. He discovered a set of conditions in terms of these symmetries that would determine if that polynomial had a solution by radicals. To do this, he worked backwards. Instead of starting with a polynomial and trying to find its roots, he started with a set of roots and looked to find the polynomial that they would form.
For example, we can use the roots -1, -2, -3, -5, and -7 to construct a polynomial and find its coefficients. To do this, we can recognize that these roots correspond to the following binomial factors:
(x+1)(x+2)(x+3)(x+5)(x+7)
When multiplied together, these factors produce the quintic polynomial expression:
x^{5} + 18x^{4} + 118x^{3} + 348x^{2} + 457x + 210
Galois studied the conditions under which the coefficients and roots were related in such a way as to permit a solution by radicals. To do this, he started with a set of roots and combined them with the rational numbers to serve as basic building blocks for creating other, somewhat arbitrary, numbers using multiplication and addition. Using these blocks, he identified a series of key equations relating the roots and examined how those equations behaved when the roots were permuted—i.e., shuffled in a way similar to the equations and graphs we saw in the previous section.
Galois discovered that the ability to solve a polynomial built out of a given set of roots depends on the invariance under permutation of the roots of those key equations, created from the roots. In essence, he saw that certain symmetries in a polynomial's roots determined whether or not that polynomial has a nice solution.
Galois not only resolved one of the great challenges of his day, he discovered an important connection between symmetry, permutation groups, and solvability of equations. Galois' discovery is perhaps one of the more unexpected results in mathematics and shows yet again how group theory provides a way to see past the superficial in order to find underlying connections. It should come as no surprise then that group theory has played an important role in understanding not only the artistic, gambling, and mathematical worlds, but also many underlying connections inherent in the physical world.
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