The set of all algebraic numbers forms a field.
This field contains all of the rational numbers, but it is a quadratically closed field.
Like the rationals, this field also has the same cardinality as the natural numbers, because we can specify and enumerate each of its members by a fixed number of integers from the polynomial equation that defines them. So it is a bit like the rationals, but we use potentially arbitrary numbers of integers to specify each number (polynomial coefficients + index of which root we are talking about) instead of just always two as for the rationals.
Each algebraic number also has a degree associated to it, i.e. the degree of the polynomial used to define it.
TODO understand.
Sometimes mathematicians go a little overboard with their naming.
Open as of 2020:
Video 1. Why π^π^π^π could be an integer by Stand-up Maths (2021) Source. Sponsored by Jane Street. Shame.
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.
Figure 1. Source.
Direct consequence of Euclid's formula.
A generalization of the Pythagorean triple infinity question.
Video 1. Beauty Is Suffering. Source.
Once you hear about the uncomputability of such problems, it makes you see that all Diophantine equation questions risk being undecidable, though in some simpler cases we manage to come up with answers. The feeling is similar to watching people trying to solve the Halting problem, e.g. in the effort to determine BB(5).
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 fields other than the integers.
And when it can't, attempt to classify which subset of the integers can be reached. E.g. Legendre's three-square theorem.
4 squares are sufficient by Lagrange's four-square theorem.
The subsets reachable with 2 and 3 squares are fully characterized by Legendre's three-square theorem and
Compared to Waring's problem, this is potentially much harder, as we can go infinitely negative in our attempts, there isn't a bound on how many tries we can have for each number.
In other words, it is unlikely to have a Conjecture reduction to a halting problem.
Video 1. 3 as the sum of the 3 cubes by Numberphile (2019) Source.
It is exactly what you'd expect from the name, Waring was watching Netflix with Goldbach, when they suddenly came up with this.
A polynomial of degree 1, i.e. of form .
A polynomial with multiple input arguments, e.g. with two inputs and :
as opposed to a polynomial with a single argument e.g. one with just :
By default, we think of polynomials over the real numbers or complex numbers.
However, a polynomial can be defined over any other field just as well, the most notable example being that of a polynomial over a finite field.
For example, given the finite field of order 9, and with elements , we can denote polynomials over that ring as
where is the variable name.
For example, one such polynomial could be:
and another one:
Note how all the coefficients are members of the finite field we chose.
Given this, we could evaluate the polynomial for any element of the field, e.g.:
and so on.
We can also add polynomials as usual over the field:
and multiplication works analogously.
The usual definition of a polynomial is over a field as shown at polynomial over a field.
However, there is nothing in the immediate definition that prevents us from having a ring instead, i.e. a field but without the commutative property and inverse elements.
The only thing is that then we would need to differentiate between different orderings of the terms of multivariate polynomial, e.g. the following would all be potentially different terms:
while for a field they would all go into a single term:
so when considering a polynomial over a ring we end up with a lot more more possible terms.
The polynomials together with polynomial addition and multiplication form a commutative ring.