The p-adic numbers are a completion of rational numbers with respect to the p-adic norm and form a non-Archimedean field. They stand in contrast to the real numbers, which are a completion of the rationals with respect to absolute value and which do respect the Archimedean property. The p-adic numbers were first described in 1987 by Kurt Hensel and later generalized in the early 20th century by József Kürschák which paved the way for a myriad of mathematical work involving the P-adics in the just over a century since.
To start talking about the p-adics it makes sense to start with the Archimedean property, which as mentioned above the p-adics do not adhere to. First we need an absolute value function or valuation, ||, which is basically a function that gives some concept of magnitude to an element of a field (so it’s a mapping from a field to the positive real numbers). Given this, the zero element should map to zero and the valuation of the product of two elements should be the same as the product of the valuation of each element individually. The last condition (which determines whether or not the valuation is Archimedean) is that if the norm of an element is less than or equal to 1, then the norm of 1 plus that element is less than some constant, C, which is independent of the choice of x. If the last condition holds for C equal to 1, then the valuation satisfies the ultrametric inequality: the valuation of 2 elements is less than or equal to the maximum of the valuation of each element. If the ultrametric inequality is satisfied then the valuation is non-Archimedean. Otherwise it is an Archimedean valuation.
While this is a bit opaque, it makes more sense now moving into defining Archimedean fields: a field is Archimedean if given a field with an associated valuation and a non-zero element, x, of that field, then there exists some natural number n such that |Σk=1nx|>1. Otherwise, the field is non-Archimedean and the ultrametric inequality holds. Basically what this means is that if we can measure distance and we are living in a nice Archimedean world, if we walk forward we can go as far as we want. While if we were to live in a non-Archimedean world and we try to walk forward we would at best stay in place and possibly move backward.
Now that that’s out of the way and (hopefully) the weirdness of a non-Archimedean world has been established, it’s time to talk about the p-adics. Any non-zero rational number, x, may be expressed in the form x ,where a and b are relatively prime to some fixed prime p and r is an integer. Using this, the p-adic norm of x is defined as |x|=p-r , which is non-Archimedean. For example, when p=3, |6|=|2*3|=1/3, |9|=|32|=1/9 and |6+9|=|15|=|3*5|=1/3 or when p=5 , |4/75|=|4/(52*3)|= 25, |13/250|=|13/(2*53)|=125 while |4/75 + 13/250|=|17/325|=|17/(52*13)|=25. So now that we have this we can proceed identically as when constructing the real numbers using the absolute value and define Qp as the set of equivalence classes of Cauchy sequences with respect the p-adic norm. After some work it can be shown that every element in Qpcan be written uniquely as Σk=makpk,where am does not equal zero and m may be any integer.
The most common use of p-adics I found was in showing the existence (or lack thereof) of rational or integer solutions to problems. For example, the Hasse principle (also known as the local-global prinicipal ) was discovered by Helmut Hasse in the 1920’s and attempts to give a partial converse of the statement that if a polynomial of the form Σaijxiyj+Σbixi+c=0 has a rational solution then it has a solution for all expansions of Q. The Hasse principal asserts that if such a polynomial has a solution in R and every Qp then it has solution in Q. An example of this is x2-2=0, which has (irrational) solution square root of 2 in R. However, it does not have solution in Q5 , and so by the Hasse principal it does not have a solution in Q, which we know to be true. Another use of the P-adics which is fairly interesting is in transferring standard real or complex polynomial equations to their tropical (the semi ring of the reals with the addition of an infinity element under the laws of composition addition and min (or max)) polynomial counterpart, a process which runs into issues due to the necessity of the ultrametric inequality.