Ciro Santilli @cirosantilli 34

The size of a set.

For finite sizes, the definition is simple, and the intuitive name "size" matches well.

But for infinity, things are messier, e.g. the size of the real numbers is strictly larger than the size of the integers as shown by Cantor's diagonal argument, which is kind of what justifies a fancier word "cardinality" to distinguish it from the more normal word "size".

The key idea is to compare set sizes with bijections.

Polynomial (possibly a multivariate polynomial) with integer coefficients.

Sometimes systems of Diophantine equations are considered.

Problems generally involve finding integer solutions to the equations, notably determining if any solution exists, and if infinitely solutions exist.

The general problem is known to be undecidable: Hilbert's tenth problem.

The Pythagorean triples, and its generalization Fermat's last theorem, are the quintessential examples.

A convenient notation for the elements of $GF(n)$ of prime order is to use integers, e.g. for $GF(7)$ we could write:which makes it clear what is the additive inverse of each element, although sometimes a notation starting from 0 is also used:

For fields of prime order, regular modular arithmetic works as the field operation.

For non-prime order, we see that modular arithmetic does not work because the divisors have no inverse. E.g. at order 6, 2 and 3 have no inverse, e.g. for 2:we see that things wrap around perfecly, and 1 is never reached.

For non-prime prime power orders however, we can find a way, see finite field of non-prime order.

The "greatest common divisor" of two integers $x$ and $y$, denoted $\text{gdc}(x,y)$ is the largest natural number that divides both of the integers.

For example, $\text{gdc}(8,12)$ is 4, because:

mathoverflow.net/questions/11540/what-are-the-most-attractive-turing-undecidable-problems-in-mathematics/11557#11557 contains a good overview of the decidability status of variants over rings other than the integers.

This section is about functions that operate on numbers such as the integers or real numbers.

Unlike over non-commutative rings, polynomials do look like proper polynomials over commutative ring.

In particular, Hilbert's tenth problem is about polynomials over the integers, which is a commutative ring, and therefore brings mindshare to this definition.

Same as recursive language but in the context of the integers.

A "reduced fraction" is a fraction that has the smallest possible integer numerator and denominator for its value.

For example:is not a reduced fraction, because there is another fraction equal to it but with smaller numerator and denominator:

A Ring can be seen as a generalization of a field where:

Addition however has to be commutative and have inverses, i.e. it is an Abelian group.

The simplest example of a ring which is not a full fledged field and with commutative multiplication are the integers. Notably, no inverses exist except for the identity itself and -1. E.g. the inverse of 2 would be 1/2 which is not in the set. More specifically, the integers are a commutative ring.

A polynomial ring is another example with the same properties as the integers.

The simplest non-commutative, non-division is is the set of all 2x2 matrices of real numbers:Note that $GL(n)$ is not a ring because you can by addition reach the zero matrix.

And when it can't, attempt to classify which subset of the integers can be reached. E.g. Legendre's three-square theorem.