Suppose I had the quintic equation $(x+1)(x+2)(x+3)(x+4)(x+5)=0$. Does the insolvability theory mean that I can only get approximations because the root is in general an irrational number, or does it mean that even in this case I can only get an approximation to say -5 as well even though it is whole number and is an exact root?
I've read that Galois theory can tell you if a quintic polynomial is the type that can be solved exactly. My question is: Are these types of quintics simply the ones with integer roots? I would suspect that just because a quintic has integer coefficients doesn't mean they have integer roots.
Finally, if Galois theory says that a particular quintic is of the exactly solvable type, what method is used to solve them exactly, besides the rational root theorem? I'm pretty sure it would be something like Cardano's method, adapted to a fifth degree equation. Where can you find this method?
Best Answer
Galois Theory assigns, to each polynomial, a mathematical structure called a group. A polynomial is solvable in radicals (that is, you can write down its roots in terms of its coefficients, the 4 arithmetical operations, and square roots, cube roots, etc.) if and only if the corresponding group is a "solvable" group. The definition of solvable group won't mean much to you if you haven't done a course in group theory; there should be a sequence of groups, starting with the trivial group and ending with the group corresponding to the polynomial, such that each group in the sequence is a "normal" subgroup of the next group, and the "quotient" of each group by the previous group is "commutative".
What's more, if the group corresponding to the polynomial is solvable, then this sequence of in-between groups, together with the quotient groups, can be used to construct the formula for the roots of the polynomial.
To expand this into something you could actually use to determine whether a polynomial is solvable in radicals, and to solve it if it is, takes a semester of advanced undergraduate level mathematics. Get yourself a good text on Galois Theory (assuming you have already done courses on Linear Algebra and an introductory Abstract Algebra course - if not, you'll have to study those first), read it, and enjoy.