Why Quintic (and Higher) Polynomials Have No Algebraic Solution

2008.03.27   prev     next

Polynomial equations of orders two, three, and four have well-known algebraic solutions. But orders five and up have been proven to have no algebraic solution. Why is that? Everywhere I’ve seen this subject discussed, it is always treated as a strange, interesting discovery. Wikipedia (as of this writing) calls it “striking.” Nowhere have I found an explanation of why this should be. So — if you’ve ever hungered for such an explanation, hunger no more. Here it is. But be warned; it’s actually pretty mundane.

If we can’t solve a fifth-order polynomial with algebra, is there any way to solve it? Sure there is — it’s called iterative approximation. Just write a computer program to try some values, and use an iterating, “honing-in” technique to get closer and closer to the correct answer. (You’ll need to run your procedure again to find each of the other answers, since there are multiple solutions.) In a very reasonable number of iterations (say, a few dozen), you can get an approximate answer that’s as accurate as you’d ever need it to be in this universe. So although fifth-order polynomials can’t be solved with algebra, they can be solved via an iterative procedure.

Now — take a hard look at this formula, the good old quadratic equation that solves second-order polynomials:

What is that “sqrt” thing? It’s shorthand for an iterative, honing-in procedure. Numbers can be straightforwardly added together, but there is no straightforward way to obtain a square root. How does your computer’s “sqrt” function do it? You guessed it — it uses an iterative procedure that gets closer and closer to the answer until the limit of the computer’s floating-point accuracy is reached.

The only reason there’s an “algebraic” solution for second-order polynomials is because we include, in our algebraic notation, a symbol for the square root procedure. Add to your algebra some new symbols — shorthand for other iterative, honing-in procedures — and you’ll probably start finding “algebraic” solutions to fifth-order and higher polynomials.

Oh and by the way, division (“/”) also requires an iterative approximation (heavily assisted by big tables in modern processors). So even a first-order polynomial needs a shorthand symbol (for an iterative procedure) in its algebraic solution.

The absence of algebraic solutions to fifth-order and higher polynomials is a not a feature of mathematics, but instead is a consequence of the arbitrary human decision to represent some iterative, approximating procedures with shorthand symbols, but not others.

That’s it. Ho, hum.