Galois theory
A polynomial equation is an equation where you are only allowed to add and subtract multiples of powers of a variable, like . An nth power is a variable multiplied by itself n times. Or example polynomial has 5th, 3rd, 2nd, and 1st powers of x. The degree of a polynomial is the largest power found in the equation. In school, students learn about a type of polynomial equation called a quadratic equation. This is an equation where the highest power, or multiple, is two. An example of a quadratic equation is, or . Notice that the only thing we care about is that there is a squared term. This makes the equation quadratic. If  are numbers and , then  is a valid quadratic equation. You may remember from school that if you know  then you can solve this equation using the quadratic formula,which gives the solutions to our equation only by plugging in . Mathematicians hoped to find a simple equation for the solutions (or, roots) for any polynomial equation, hopefully using only addition, subtraction, multiplication, division, and two other simple calculations: powers and roots. Mathematicians were able, after years of work, to find equations for polynomials with degree 3, and eventually degree 4. However, no one was ever able to find a formula for the solutions to a polynomial with degree 5 or higher. Why not? What was preventing them from being able to plug in the (in this case, since there are 5 powers)  in  and get the answers? No one could find out why.