Galois Theory and the Solubility of Polynomials by Radicals
Given a polynomialwith rational coefficients, can you express the roots of p(x) using only rational numbers, multiplication, division, addition, subtraction and taking integer roots? So, for example, we can solvein this way because
The coefficientsare rational, and we have only used multiplication, division, addition, subtraction and square root.
We can find more complicated examples, suppose p(x)=x^4 +4x^2+2. We can write this asso the solutions will satisfy The roots are
When we can find the solutions for a polynomial with rational coefficients using only rational numbers and the operations of addition, subtraction, division, multiplication and finding nth roots, we say thatis soluble by radicals.
Using Galois theory, we can prove that if the degree of(the highest power ofin) is less than 5 then the polynomial is soluble by radicals, but there are polynomials of degree 5 and higher not soluble by radicals. In other words, polynomials of degree 5 whose solutions cannot be written down using nth roots and the arithmetical operations, no matter how complicated.
We can construct a group to act of the set of roots of a polynomial – called a group action. Such a group will be an automorphism of the roots. For example the group acting on the roots of the polynomialare
For a polynomial of degreethe group will be a subgroup ofThe group generated will have subgroups which may or may not be normal inIf the subgroup is normal inthe the polynomial is soluble by radicals else it is not. Forandall the subgoups are normal butandforhas subgroups which are not normal, so polynomials of degree 5 or greater are not soluble by radicals in general although some may be.