## Galois Theory and the Solubility of Polynomials by Radicals

Given a polynomial with 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 solve in this way because The coefficients are 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 as so 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 that is soluble by radicals.

Using Galois theory, we can prove that if the degree of (the highest power of in ) 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 polynomial are and For a polynomial of degree the group will be a subgroup of The group generated will have subgroups which may or may not be normal in If the subgroup is normal in the the polynomial is soluble by radicals else it is not. For and all the subgoups are normal but and for has subgroups which are not normal, so polynomials of degree 5 or greater are not soluble by radicals in general although some may be. 