This chart shows how to solve every quartic polynomial.

Notice it also shows how to solve cubics and quadratics, but it can’t be improved to quintics (polynomials of degree five) or higher, because only polynomials with degree less than five can be solved algebraically in general: this is the Abel-Ruffini theorem. Some specific quintics can be solved, but the method is far more tedious. In 2004 Daniel Lazard wrote out a three-page formula for the roots of a general solvable quintic.

Another thing I’d like to add now that you’ve mentioned the Abel-Ruffini Theorem is that although Ruffini released an incomplete proof in 1799 and Abel managed to prove it completely in 1823, Galois at roughly the age of 17 managed to — independently — prove the same result. 

More astonishingly, Abel and Galois both invented an area of mathematics called Group Theory to aid in their proofs of the problem, completely independently of one another. 

Also, although Abel was the first to solve it after the problem had remained unsolved for some 250 years, and although the method was in a sense similar to that of Galois, the teenage Galois was still able to produce a proof that led to a deeper understanding of polynomials. 

(This proof also helped Galois develop Galois theory, which turned out to be incredibly helpful and applicable to a variety of area other than the theory of equations). 


Prime numbers are like atoms. They are the building blocks of all integers. Every integer is either itself a prime or the product of primes. For example, 11 is a prime; 12 is the product of the primes 2, 2, and 3; 13 is a prime; 14 is the product of the primes 2 and 7; 15 is the product of the primes 3 and 5; and so on. Some 2,300 years ago, in proposition 20 of Book IX of his Elements, Euclid gave a proof “straight from the book” that the supply of primes is inexhaustible.

Assume, said Euclid, that there is a finite number of primes. Then one of them, call it P, will be the largest. Now consider the number Q, larger than P, that is equal to the product of the consecutive whole numbers from 2 to P plus the number 1. In other words, Q = (2 x 3 x 4 … x P) + 1.

From the form of the number Q, it is obvious that no integer from 2 to P divides evenly into it; each division would leave a remainder of 1. If Q is not prime, it must be evenly divisible by some prime larger than P. On the other hand, if Q is prime, Q itself is a prime larger than P.

Either possibility implies the existence of a prime larger than the assumed largest prime P. This means, of course, that the concept of “the largest prime” is a fiction. But if there’s no such beast, the number of primes must be infinite. “Euclid alone,” wrote Edna St. Vincent Millay, “has looked upon Beauty bare.”

Paul Hoffman (and Euclid) in The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth.

if you don’t think this is the coolest fucking thing idek what to say

(via deflect)