Number theory would be unrecognizable today without the incredible contributions of Dr Emmy Noether in the early 1900s. The study of polynomial roots had exploded in the 1800s, but the new number systems broke a fundamental properties of integers -- unique factorization into prime numbers.

Since the prime numbers were flawed, Noether came up with "ideals" to take their place. An ideal is a set of numbers which behave the same under multiplication, like how we can categorize integers into evens and odds.

Noether was interested in chains of ideals, each ideal contained in one that's bigger. Number systems where these chains are finite are now called 'Noetherian domains,' in honor of her work. Besides algebra, she also made major contributions to physics and topology.

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"The Theory of Ideals in Ring Domains"
We're grateful, Emmy Noether
A woman of talent and major brains,
I'm talkin' Emmy Noether

With her vision and abstract mind,
she changed the face of algebra.
With her vision she saved the day,
when all the primes had broken down

Let x be the square root of minus five,
Imaginary number
It ruins your factors and that don't jive,
Probationary number

Sets of numbers with simple rules,
will multiply and draw you in
Sets of numbers we call ideals
will take the place of factoring

Make a new ideal
Numbers all together
Make a new ideal
Thank you Emmy Noether

Once in an ideal you can't get out,
They get you with the times sign
But that is their strong suit and there's no doubt
We're gonna need that times sign

Number systems with simple rules,
use their ideals for factoring
Number systems we give the name
"Noetherian" to honor her

Make a new ideal
Numbers all together
Make a new ideal
Thank you Emmy Noether

Taking out an ideal
And putting it in another
Climb the ascending chain
We have to stop here says Noether

Taking out an ideal
And putting it in another
Climb the ascending chain
We have to stop here says Noether

The people in physics say give to you,
They're grateful, Emmy Noether
You helped David Hilbert, and Einstein, too!
So thank you, Emmy Noether

With your passion and abstract mind,
you changed the face of algebra.
With your passion you saved the day,
when all the primes had broken down

Make a new ideal
Numbers all together
Make a new ideal
Thank you Emmy Noether