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# algebra

## Latest Articles

### The Simple Math Behind the Mighty Roots of Unity

Solutions to the simplest polynomial equations — called “roots of unity” — have an elegant structure that mathematicians still use to study some of math’s greatest open questions.

### How to Find Rational Points Like Your Job Depends on It

Using high school algebra and geometry, and knowing just one rational point on a circle or elliptic curve, we can locate infinitely many others.

### Mathematician Disproves 80-Year-Old Algebra Conjecture

Inside the symmetries of a crystal shape, a postdoctoral researcher has unearthed a counterexample to a basic conjecture about multiplicative inverses.

### Mathematicians Resurrect Hilbert’s 13th Problem

Long considered solved, David Hilbert’s question about seventh-degree polynomials is leading researchers to a new web of mathematical connections.

### When Math Gets Impossibly Hard

Mathematicians have long grappled with the reality that some problems just don’t have solutions.

### The ‘Useless’ Perspective That Transformed Mathematics

Representation theory was initially dismissed. Today, it’s central to much of mathematics.

### The Map of Mathematics

Explore our surprisingly simple, absurdly ambitious and necessarily incomplete guide to the boundless mathematical universe.

### Mathematicians Cut Apart Shapes to Find Pieces of Equations

New work on the problem of “scissors congruence” explains when it’s possible to slice up one shape and reassemble it as another.

### The (Imaginary) Numbers at the Edge of Reality

Odd enough to potentially model the strangeness of the physical world, complex numbers with “imaginary” components are rooted in the familiar.