7

So according to Abel-Ruffini Theorem, it states that there is no algebraic solution, in the form of radicals, to general polynomials of degree $5$ or higher.

But I'm wondering if there is a way to decide whether a polynomial, such as $$x^5+14x^4+12x^3+9x+2=0$$ has roots that can be expressed in radicals or not just by having a glance at the polynomial.

Frank
  • 5,984
  • 1
    You could compute its Galois Group to check if you can solve it in radicals. – Edward Evans Oct 06 '16 at 23:08
  • @Ed_4434 What's a "Galois Group"? I have some knowledge in group theory, but I have never heard the term Galois Group. Is it something you commonly learn in school? – Frank Oct 06 '16 at 23:09
  • 2
    Try Wikipedia. Of course that requires a lot more than a glance. A CAS would be handy. – Robert Israel Oct 06 '16 at 23:11
  • 3
    Modulo $3$ it has a factor of degree $3$, and modulo $7$ it is irreducible, so its Galois group contains $A_5$, hence it is not solvable. That's a bit more than a glance though, and requires some knowledge of group theory. – Servaes Oct 06 '16 at 23:12
  • @Frank It's a more or less advanced undergraduate/graduate object which in some sense looks at the solvability of a polynomial. – Edward Evans Oct 06 '16 at 23:13
  • 2
    +1 for the good question. Unfortunately, as commenters tell you, there is no way to answer it given your level of mathematical knowledge. – Ethan Bolker Oct 06 '16 at 23:23
  • A course on Galois Theory tends to lead you through the computation of galois groups and related concepts. The "punchline" of the course is usually a proof of Abel-Ruffini which is definitely something to aim for if you're interested in algebra. – Edward Evans Oct 06 '16 at 23:31
  • Always check the rational roots theorem first? – Simply Beautiful Art Oct 07 '16 at 01:16
  • 1
    @Simple Art Yes. But none of the roots work. The problem with the rational roots theorem is if the root is... well, irrational. Like $\sqrt2$. So just the roots from the rational root theorem isn't going to decide whether the polynomial can be solved. – Frank Oct 07 '16 at 02:50
  • @Ethan Bolker Oh.. Then what do I have to learn to be able to decide whether or not a polynomial can be solvable? I see alot of stuff about groups, so I'm gonna guess group theory... is that it? – Frank Oct 07 '16 at 02:53
  • @Frank Well, it doesn't hurt to check the rational roots theorem. And I'm not too familiar with the stuff, but you should learn Galois theory, as mentioned, to determine if it is factorable. – Simply Beautiful Art Oct 07 '16 at 12:28
  • As @Ed_4434 comments, you'll need to get as far in abstract algebra as Galois theory. That's at least one course, probably two. Whether you can do it by reading depends on how firm your "some knowledge of group theory" is, and your personality and determination. If you want to read, consider asking a separate question and making clear your current state of knowledge. – Ethan Bolker Oct 07 '16 at 12:32

1 Answers1

4

As the others have commented, to know when a quintic (or higher) is solvable in radicals requires Galois theory. However, there is a rather simple aspect when it is not solvable that is easily understood and can be used as a litmus test.

Theorem: An irreducible equation of prime degree $p>2$ that is solvable in radicals has either $1$ or $p$ real roots.

(Irreducible, simply put, means it has no rational roots.) By sheer coincidence, the irreducible quintic you chose has $3$ real roots so, by looking at its graph, you can indeed tell at a glance that this is not solvable in radicals. Going higher, if an irreducible septic has $3$ or $5$ real roots, then you automatically know it is not solvable. And so on.

P.S. And before you ask, it does not work the other direction: if it has $1$ or $p$ real roots, it does not imply it is solvable in radicals. It is a necessary but not sufficient condition.

Tito Piezas III
  • 54,565
  • I have studied Galois theory, but I have never come across that theorem! How is it proved? – Unit Nov 03 '16 at 15:20
  • 1
    @Unit: I believe Dummit mentions it briefly in his paper on quintics, but I know it was established before him. How to prove it, I leave to those more capable than I. :) – Tito Piezas III Nov 03 '16 at 15:24