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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク P-adic number ： ウィキペディア英語版 P-adic number In mathematics the -adic number system for any prime number  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 extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, -adic numbers have the interesting property that they are said to be close when their difference is divisible by a high power of – the higher the power the closer they are. This property enables -adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles.〔F. Q. Gouvêa, A Marvelous Proof, The American Mathematical Monthly, Vol. 101, No. 3 (Mar., 1994), pp. 203–222〕 -adic numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Kummer's earlier work can be interpreted as implicitly using -adic numbers.〔. Translation into English by John Stillwell of ''Theorie der algebraischen Functionen einer Veränderlichen'' (1882). Translator's introduction, (page 35 ): "Indeed, with hindsight it becomes apparent that a discrete valuation is behind Kummer's concept of ideal numbers."〕 The -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 essentially provides an alternative form of calculus. More formally, for a given prime , the field Q''p'' of -adic numbers is a completion of the rational numbers. The field Q''p'' is also given a topology derived from a metric, which is itself derived from the p-adic order, an alternative valuation on the rational numbers. This metric space is complete in the sense that every Cauchy sequence converges to a point in Q''p''. 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 -adic number systems their power and utility. The in "-adic" is a variable and may be replaced with a prime (yielding, for instance, "the 2-adic numbers") or another ''placeholder variable'' (for expressions such as "the ℓ-adic numbers"). The "adic" of "-adic" comes from the ending found in words such as dyadic or triadic, and the means a prime number. ==Introduction== ''This section is an informal introduction to p-adic numbers, using examples from the ring of 10-adic (decadic) numbers. Although for p-adic numbers p should be a prime, base 10 was chosen to highlight the analogy with decimals. The decadic numbers are generally not used in mathematics: since 10 is not prime, the decadics are not a field. More formal constructions and properties are given below.'' In the standard decimal representation, almost all〔The number of real numbers with terminating decimal representations is countably infinite, while the number of real numbers without such a representation is uncountably infinite.〕 real numbers do not have a terminating decimal representation. For example, 1/3 is represented as a non-terminating decimal as follows :$\backslash frac=0.333333\backslash ldots.$ Informally, non-terminating decimals are easily understood, because it is clear that a real number can be approximated to any required degree of precision by a terminating decimal. 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". Whereas two decimal expansions are close to one another if their difference is a large negative power of 10, two 10-adic expansions are close if their difference is a large positive power of 10. Thus 3333 and 4333, which differ by 103, are close in the 10-adic world, and 33333333 and 43333333 are even closer, differing by 107. More precisely, a positive rational number  can be expressed as , where and are positive integers and is relatively prime to and to 10. For each there exists the maximal  such that this representation is possible. Let the 10-adic absolute value of  to be :       $|r|\_\; =\; \backslash frac$ :       |0|10 = 0. Closeness in any number system is defined by a metric. Using the 10-adic metric the distance between numbers and is given by . An interesting consequence of the 10-adic metric (or of a -adic metric) is that there is no longer a need for the negative sign. As an example, by examining the following sequence we can see how unsigned 10-adics can get progressively closer and closer to the number −1: :$9=-1+10\; \backslash ,$        so  $|9-(-1)|\_\; =\; \backslash frac$. :$99=-1+10^2\; \backslash ,$       so  $|99-(-1)|\_\; =\; \backslash frac$. :$999=-1+10^3\; \backslash ,$       so  $|999-(-1)|\_\; =\; \backslash frac$. :$9999=-1+10^4\; \backslash ,$       so  $|9999-(-1)|\_\; =\; \backslash frac$. and taking this sequence to its limit, we can say that the 10-adic expansion of −1 is :$\backslash dots\; 9999=-1.\backslash ,$ 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 -adic numbers – for alternatives see the ''Notation'' section below. More formally, a 10-adic number can be defined as :$\backslash sum\_^\backslash infty\; a\_i\; 10^i$ where each of the ''a''''i'' is a digit taken from the set and the initial index may be positive, negative or 0, but must be finite. From this definition, it is clear that positive integers and positive rational numbers 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 :$-100\; =\; -1\; \backslash times\; 100\; =\; \backslash dots\; 9999\; \backslash times\; 100\; =\; \backslash dots\; 9900\; \backslash ,$ :$\backslash Rightarrow\; -35\; =\; -100+65\; =\; \backslash dots\; 9900\; +\; 65\; =\; \backslash dots\; 9965\; \backslash ,$ :$\backslash Rightarrow\; -\backslash left(3+\backslash dfrac\backslash right)=\backslash dfrac=\; \backslash dfrac=\backslash dots\; 9996.5$ and fractions which have non-terminating decimal expansions also have non-terminating 10-adic expansions. For example :$\backslash dfrac=142857;$ \dfrac=142857142857; \dfrac=142857142857142857 :$\backslash Rightarrow-\backslash dfrac=\backslash dots\; 142857142857142857$ :$\backslash Rightarrow-\backslash dfrac=\backslash dots\; 142857142857142857\; \backslash times\; 6\; =\; \backslash dots\; 857142857142857142$ :$\backslash Rightarrow\backslash dfrac\; =\; -\backslash dfrac+1\; =\; \backslash dots\; 857142857142857143.$ Generalizing the last example, we can find a 10-adic expansion with no digits to the right of the decimal point for any rational number such that is co-prime to 10; Euler's theorem guarantees that if is co-prime to 10, then there is an such that is a multiple of . The other rational numbers can be expressed as 10-adic numbers with some digits after the decimal point. As noted above, 10-adic numbers have a major drawback. It is possible to find pairs of non-zero 10-adic numbers (having an infinite number of digits, and thus not rational) whose product is 0.〔(See Gérard Michon's article at )〕 This means that 10-adic numbers do not always have multiplicative inverses i.e. valid reciprocals, which in turn implies that though 10-adic numbers form a ring they do not form a field, a deficiency that makes them much less useful as an analytical tool. Another way of saying this is that the ring of 10-adic numbers is not an integral domain because they contain zero divisors. The reason for this property turns out to be that 10 is a composite number which is not a power of a prime. This problem is simply avoided by using a prime number  as the base of the number system instead of 10 and indeed for this reason in -adic is usually taken to be prime. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「P-adic number」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.