What were Niels Henrik Abel's contributions?

Answer 1

Abel proved, at the age of 19 mind you, that there is no general algebraic solution to polynomial equations of degree 5 or higher, as opposed to say, the quadratic equation for a 2nd degree polynomial. At that time, that was one of the biggest unsolved problems in math, so as you might imagine, proving it was a big deal.

However, as it turns out, the far more important result from this proof was that Abel had to invent the algebraic subfield known as group theory to complete it, and group theory provides the current mathematical basis to our model of fundamental particles and forces, known as the standard model.