P-adic Field

In Mathematics , the ''p''-adic number systems were first described by Kurt Hensel in 1897 . For each Prime Number ''p'', the ''p''-adic Number System extends the ordinary Arithmetic of the Rational Numbers in a way different from the extension of the rational number system to the Real and Complex number systems. The main use of these other systems is in Number Theory . The extension is achieved by an alternative interpretation of the concept of Absolute Value . The ''p''-adic numbers were motivated primarily by an attempt to bring the ideas and techniques of Power Series methods into number theory. Their influence now extends far beyond this. For example, the field of ''p''-adic Analysis essentially provides an alternative form of Calculus . More formally, for a given prime ''p'', the Field Q_''p'' of ''p''-adic numbers is a Completion of the Rational Number s. The field Q_''p'' is also given a Topology derived from a Metric , which is itself derived from an alternative Valuation on the rational numbers. This metric space is Complete in the sense that every Cauchy Sequence converges. This is what allows the development of calculus on Q_''p'', and it is the interaction of this analytic and algebraic structure which gives the ''p''-adic number systems their power and utility. The ''p'' in ''p''-adic is a ''dummy variable.'' Advanced articles in number theory often speak of the ''l''-adic numbers without explanation. The ''l''-adic numbers are the same thing as the ''p''-adic numbers. INTRODUCTION ''This section is an informal introduction to p-adic numbers, using examples from the ring of 10-adic numbers. More formal constructions and properties are given below.'' In the standard Decimal Representation , many (in fact, most) Real Number s do not have a terminating decimal expansion. For example, 1/3 is represented as a non-terminating decimal as follows : $rac{1}{3}=0.333333\dots$ Informally, most people are comfortable with non-terminating decimals because it is clear that a real number can be approximated to any required degree of closeness by a terminating decimal that uses enough decimal places. If two decimal expansions differ only after the 10th decimal place they are quite close to one another, and if they differ only after the 20th decimal place they are even closer. 10-adic numbers use a similar non-terminating expansion, but with a different concept of "closeness" (which mathematicians call a Metric ). Whereas two decimal expansions are close to one another if they differ by a large negative power of 10, two 10-adic expansions are close if they differ by a large positive power of 10. Thus 3333 and 4333 are close in the 10-adic metric, and 33333333 and 43333333 are even closer. In the 10-adic metric, the following sequence of numbers gets closer and closer to −1 : $9=-1+10$ : $99=-1+10^2$ : $999=-1+10^3$ : $9999=-1+10^4$ and taking this sequence to its limit, we can say (informally) that the 10-adic expansion of −1 is : $\dots 9999=-1\,$ In this notation, 10-adic expansions can be extended indefinitely to the left, in contrast to decimal expansions, which can be extended indefinitely to the right. Note that this is not the only way to write ''p''-adic numbers—for alternatives see the ''Notation'' section below. More formally, a 10-adic number can be defined as : $\sum_{i=n}^\infty a_i 10^i$ where each of the ''a''_''i'' is a Digit taken from the set {0, 1, …..., 9} and the initial index ''n'' may be positive, negative or 0, but must be finite. From this definition, it is clear that positive integers and positive Rational Number s with terminating decimal expansions will have terminating 10-adic expansions that are identical to their decimal expansions. Other numbers may have non-terminating 10-adic expansions. It is possible to define addition, subtraction, and multiplication on 10-adic numbers in a consistent way, so that the 10-adic numbers form a Commutative Ring . We can create 10-adic expansions for negative numbers as follows : $-100 = -1 imes 100 = \dots 9999 imes 100 = \dots 9900$ : $\Rightarrow -35 = -100+65 = \dots 9900 + 65 = \dots 9965$ : $\Rightarrow -3rac{1}{2}=rac{-35}{10}= rac{\dots 9965}{10}=\dots 9996.5$ and fractions which have non-terminating decimal expansions also have non-terminating 10-adic expansions. For example : $rac{10^6-1}{7}=142857; rac{10^{12}-1}{7}=142857142857; rac{10^{18}-1}{7}=142857142857142857$ : $\Rightarrow-rac{1}{7}=\dots 142857142857142857$ : $\Rightarrow-rac{6}{7}=\dots 142857142857142857 imes 6 = \dots 857142857142857142$ : $\Rightarrowrac{1}{7} = -rac{6}{7}+1 = \dots 857142857142857143$ Generalizing the last example, we can find a 10-adic expansion for any rational number ''p''⁄''q'' such that ''q'' is co-prime to 10; Euler's Theorem guarantees that if ''q'' is co-prime to 10, then there is an ''n'' such that 10^''n'' − 1 is a multiple of ''q''. However, 10-adic numbers have one major drawback. It is possible to find pairs of non-zero 10-adic numbers whose product is 0. In other words, the 10-adic numbers are not a Domain because they contain Zero Divisor s. This turns out to be because 10 is a Composite number. Fortunately, this problem can be avoided by using a prime number ''p'' as the Base of the number system instead of 10. P-ADIC EXPANSIONS If ''p'' is a fixed prime number, then any positive Integer can be written in a Base p expansion in the form : $\sum_{i=0}^n a_i p^i$ where the a_i are integers in {0, …, ''p'' − 1}. For example, the Binary expansion of 35 is 1·2⁵ + 0·2⁴ + 0·2³ + 0·2² + 1·2¹ + 1·2⁰, often written in the shorthand notation 100011₂. The familiar approach to generalizing this description to the larger domain of the rationals (and, ultimately, to the reals) is to include sums of the form: : $\pm\sum_{i=-\infty}^n a_i p^i$ A definite meaning is given to these sums based on Cauchy Sequence s, using the Absolute Value as metric. Thus, for example, 1/3 can be expressed in base 5 as the limit of the sequence 0.1313131313...₅. In this formulation, the integers are precisely those numbers which can be represented in the form where ''a''_''i'' = 0 for all ''i'' < 0. As an alternative, if we extend the base p expansions by allowing infinite sums of the form : $\sum_{i=k}^{\infty} a_i p^i$ where ''k'' is some (not necessarily positive) integer, we obtain the ''p''-adic expansions defining the of integers modulo ''p''. If each ring is needed, the latter is usually written Z/''p''Z or Z/''(p)''. Be sure to check the notation for any text you read.) Intuitively, as opposed to ''p''-adic expansions which extend to the ''right'' as sums of ever smaller, increasingly negative powers of the base ''p'' (as is done for the real numbers as described above), these are numbers whose ''p''-adic expansion to the ''left'' are allowed to go on forever. For example, the ''p''-adic expansion of 1/3 in base 5 is …1313132, i.e. the limit of the sequence 2, 32, 132, 3132, 13132, 313132, 1313132,… . Multiplying this infinite sum by 3 in base 5 gives …0000001. As there are no negative powers of 5 in this expansion of 1/3 (i.e. no numbers to the right of the decimal point), we see that 1/3 is a ''p''-adic integer in base 5. While it is possible to use this approach to rigorously define p-adic numbers and explore their properties, just as in the case of real numbers other approaches are generally preferred. Hence we want to define a notion of infinite sum which makes these expressions meaningful, and this is most easily accomplished by the introduction of the ''p''-adic Metric . Two different but equivalent solutions to this problem are presented in the ''Constructions'' section below. NOTATION There are several different conventions for writing ''p''-adic expansions. So far this article has used a notation for ''p''-adic expansions in which Powers of ''p'' increase from right to left. With this right-to-left notation the 3-adic expansion of ¹/₅, for example, is written as : $rac{1}{5}=\dots 121012102_3$ When performing arithmetic in this notation, digits are Carried to the left. It is also possible to write ''p''-adic expansions so that the powers of ''p'' increase from left to right, and digits are carried to the right. With this left-to-right notation the 3-adic expansion of ¹/₅ is : $rac{1}{5}=0.201210121\dots_3\mbox{ or }rac{1}{15}=2.01210121\dots_3.$ ''p''-adic expansions may be written with other sets of digits instead of {0, 1, …, ''p'' − 1}. For example, the 3-adic expansion of ¹/₅ can be written using Balanced Ternary digits {1,0,1} as : $rac{1}{5}=\dots\underline{1}11\underline{11}11\underline{11}11\underline{1}_3.$ In fact any set of ''p'' integers which are in distinct residue classes Modulo ''p'' may be used as ''p''-adic digits. In number theory, Teichmüller digits are sometimes used. CONSTRUCTIONS Analytic approach The Real Number s can be defined as Equivalence Class es of Cauchy Sequence s of Rational Number s; this allows us to, for example, write 1 as 1.000… = 0.999… . However, the definition of a Cauchy sequence relies on the Metric chosen and, by choosing a different one, numbers other than the real numbers can be constructed. The usual metric which yields the real numbers is called the Euclidean Metric . For a given prime ''p'', we define the ''p-adic norm'' in Q as follows:
	:<math>x 2	2 \,\!</math>
	:<math>x 3	1/9 \,\!</math>
	:<math>x 5	25 \,\!</math>
	:<math>x 7	1/7 \,\!</math>
	:<math>x {11}	11 \,\!</math>

:<math>d P(x,y) x-y_p \,\!</math>
:''f'': '''Q'''''p'' → '''Q'''''p'', ''f''(''x'') (1/''x''''p'')2 for ''x'' ≠ 0, ''f''(0) = 0,
:<math>x P c^{-\operatorname{ord}_P(x)}</math>

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P-adic Field