Editing Prime number (section)

=== ''p''-adic numbers ===
{{main|p-adic number}}
The [[p-adic order|{{tmath|p}}-adic order]] <math>\nu_p(n)</math> of an integer {{tmath|n}} is the number of copies of {{tmath|p}} in the prime factorization of {{tmath|n}}. The same concept can be extended from integers to rational numbers by defining the {{tmath|p}}-adic order of a fraction <math>m/n</math> to be {{tmath|\nu_p(m)-\nu_p(n)}}. The {{tmath|p}}-adic absolute value <math>|q|_p</math> of any rational number {{tmath|q}} is then defined as {{tmath|1= \vert q \vert_p=p^{-\nu_p(q)} }}. Multiplying an integer by its {{tmath|p}}-adic absolute value cancels out the factors of {{tmath|p}} in its factorization, leaving only the other primes. Just as the distance between two real numbers can be measured by the absolute value of their distance, the distance between two rational numbers can be measured by their {{tmath|p}}-adic distance, the {{tmath|p}}-adic absolute value of their difference. For this definition of distance, two numbers are close together (they have a small distance) when their difference is divisible by a high power of {{tmath|p}}. In the same way that the real numbers can be formed from the rational numbers and their distances, by adding extra limiting values to form a [[complete field]], the rational numbers with the {{tmath|p}}-adic distance can be extended to a different complete field, the [[p-adic number|{{tmath|p}}-adic numbers]].<ref name="childress"/><ref>{{cite book | last1 = Erickson | first1 = Marty | last2 = Vazzana | first2 = Anthony | last3 = Garth | first3 = David | edition = 2nd | isbn = 978-1-4987-1749-6 | mr = 3468748 | page = 200 | publisher = CRC Press | location = Boca Raton, FL | series = Textbooks in Mathematics | title = Introduction to Number Theory | url = https://books.google.com/books?id=QpLwCgAAQBAJ&pg=PA200 | year = 2016}}</ref>

This picture of an order, absolute value, and complete field derived from them can be generalized to [[algebraic number field]]s and their [[Valuation (algebra)|valuations]] (certain mappings from the [[multiplicative group]] of the field to a [[totally ordered group|totally ordered additive group]], also called orders), [[Absolute value (algebra)|absolute values]] (certain multiplicative mappings from the field to the real numbers, also called [[Norm (mathematics)|norm]]s),<ref name="childress">{{cite book | last = Childress | first = Nancy | doi = 10.1007/978-0-387-72490-4 | isbn = 978-0-387-72489-8 | mr = 2462595 | pages = 8–11 | publisher = Springer, New York | series = Universitext | title = Class Field Theory | url = https://books.google.com/books?id=RYdy4PCJYosC&pg=PA8 | year = 2009 }} See also p. 64.</ref> and places (extensions to [[complete field]]s in which the given field is a [[dense set]], also called completions).<ref>{{cite book | last = Weil | first = André | author-link = André Weil | isbn = 978-3-540-58655-5 | mr = 1344916 | page = [https://archive.org/details/basicnumbertheor00weil_866/page/n56 43] | publisher = Springer-Verlag | location = Berlin | series = Classics in Mathematics | title = Basic Number Theory | url = https://archive.org/details/basicnumbertheor00weil_866 | url-access = limited | year = 1995}} Note however that some authors such as {{harvtxt|Childress|2009}} instead use "place" to mean an equivalence class of norms.</ref> The extension from the rational numbers to the [[real number]]s, for instance, is a place in which the distance between numbers is the usual [[absolute value]] of their difference. The corresponding mapping to an additive group would be the [[logarithm]] of the absolute value, although this does not meet all the requirements of a valuation. According to [[Ostrowski's theorem]], up to a natural notion of equivalence, the real numbers and {{tmath|p}}-adic numbers, with their orders and absolute values, are the only valuations, absolute values, and places on the rational numbers.<ref name="childress"/> The [[local–global principle]] allows certain problems over the rational numbers to be solved by piecing together solutions from each of their places, again underlining the importance of primes to number theory.<ref>{{cite book | last = Koch | first = H. | doi = 10.1007/978-3-642-58095-6 | isbn = 978-3-540-63003-6 | mr = 1474965 | page = 136 | publisher = Springer-Verlag | location = Berlin | title = Algebraic Number Theory | url = https://books.google.com/books?id=wt1sCQAAQBAJ&pg=PA136 | year = 1997| citeseerx = 10.1.1.309.8812 }}</ref>