Why is it called the Fundamental Theorem of Arithmetic?

$\begingroup$

The Fundamental Theorem of Arithmetic is easy enough to understand, saying that every integer greater than 1 is either prime or is the product of a unique combination of prime numbers.

What I don't understand is why this is "fundamental." This may have massively important implications in number theory and cryptography and whatever else, but in terms of arithmetic, which I think of as adding, subtracting, multiplying, and dividing, it doesn't really actually seem to have that much importance....I don't see why it should be so fundamental.

Can someone explain its importance, or why it isn't called perhaps the "Fundamental Theorem of Number Theory"?

I would expect the Fundamental Theorem of Arithmetic to be something.

$\endgroup$ 6

1 Answer

$\begingroup$

Because Arithmetic is another name for Number Theory.

Unique factorization was used widely for ages without anyone bothering to prove it or even feeling any need for a proof. It was Gauss that recognized this and finally proved it in Disquisitiones Arithmeticae in 1801.

The Fundamental Theorem of Arithmetic is also important because it does not hold in all number rings (that is, rings of integers of an algebraic number field). Attempts to understand this led to the important development of ideal numbers by Kummer and Dedekind and the birth of algebraic number theory and modern algebra.

$\endgroup$ 4

Your Answer

Sign up or log in

Sign up using Google Sign up using Facebook Sign up using Email and Password Submit

Post as a guest

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy