## English

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### Noun

1. (number theory) An element of a completion of the field of rational numbers with respect to a p-adic ultrametric.[1]
The expansion (21)2121p is equal to the rational p-adic number ${\displaystyle \textstyle {2p+1 \over p^{2}-1}.}$
In the set of 3-adic numbers, the closed ball of radius 1/3 "centered" at 1, call it B, is the set ${\displaystyle \textstyle \{x|\exists n\in \mathbb {Z} .\,x=3n+1\}.}$ This closed ball partitions into exactly three smaller closed balls of radius 1/9: ${\displaystyle \{x|\exists n\in \mathbb {Z} .\,x=1+9n\},}$ ${\displaystyle \{x|\exists n\in \mathbb {Z} .\,x=4+9n\},}$ and ${\displaystyle \{x|\exists n\in \mathbb {Z} .\,x=7+9n\}.}$ Then each of those balls partitions into exactly 3 smaller closed balls of radius 1/27, and the sub-partitioning can be continued indefinitely, in a fractal manner.
Likewise, going upwards in the hierarchy,
B is part of the closed ball of radius 1 centered at 1, namely, the set of integers. Two other closed balls of radius 1 are "centered" at 1/3 and 2/3, and all three closed balls of radius 1 form a closed ball of radius 3, ${\displaystyle \{x|\exists n\in \mathbb {Z} .\,x=1+{n \over 3}\},}$ which is one out of three closed balls forming a closed ball of radius 9, and so on.
• 1914, Bulletin of the American Mathematical Society, page 452,
3. In his recent book Professor Hensel has developed a theory of logarithms of the rational p-adic numbers, and from this he has shown how all such numbers can be written in the form ${\displaystyle p^{\alpha }\omega ^{\beta }e^{\gamma }}$.
• 1991, M. D. Missarov, Renormalization Group and Renormalization Theory in p-Adic and Adelic Scalar Models, Ya. G. Sinaĭ (editor), Dynamical Systems and Statistical Mechanics: From the Seminar on Statistical Physics held at Moscow State University, American Mathematical Society, page 143,
p-Adic numbers were introduced in mathematics by K. Hensel, and this invention led to substantial developments in number theory, where p-adic numbers are now as natural as ordinary real numbers. [] Bleher noticed in [19] that the set of purely fractional p-adic numbers is an example of hierarchical lattice.
• 2000, Masato Kuwata (translator), Kazuya Kato, Nobushige Kurokawa, Takeshi Saitō, Takeshi Saito, Number Theory: Fermat's dream, American Mathematical Society, page 58,
${\displaystyle \mathbb {Q} _{p}}$ is called the p-adic number field, and its elements are called p-adic numbers. In this section we introduce the p-adic number fields, which are very important objects in number theory.
The p-adic numbers were originally introduced by Hensel around 1900.

#### Usage notes

• An expanded, constructive definition:
• For given ${\displaystyle p}$, the natural numbers are exactly those expressible as some finite sum ${\displaystyle \textstyle \sum _{k=0}^{n}a_{k}p^{k}}$, where each ${\displaystyle a_{k}}$ is an integer: ${\displaystyle 0\leq a_{k} and ${\displaystyle n\geq 0}$. (To this extent, ${\displaystyle p}$ acts exactly like a base).
• The slightly more general sum ${\displaystyle \textstyle \sum _{k=N}^{n}a_{k}p^{k}}$ (where ${\displaystyle N}$ can be negative) expresses a class of fractions: natural numbers divided by a power of ${\displaystyle p}$.
• Much more expressiveness (to encompass all of ${\displaystyle \mathbb {Q} }$) results from permitting infinite sums: ${\displaystyle \textstyle \sum _{k=N}^{\infty }a_{k}p^{k}}$.
• The p-adic ultrametric and the limitation on coefficients together ensure convergence, meaning that infinite sums can be manipulated to produce valid results that at times seem paradoxical. (For example, a sum with positive coefficients can represent a negative rational number. In fact, the concept negative has limited meaning for p-adic numbers; it is best simply interpreted as additive inverse.)
• Forming the completion of ${\displaystyle \mathbb {Q} }$ with respect to the ultrametric means augmenting it with the limit points of all such infinite sums.
• The augmented set is denoted ${\displaystyle \mathbb {Q} _{p}}$.
• The construction works generally (for any integer ${\displaystyle p>1}$), but it is only for prime ${\displaystyle p}$ that it becomes of significant mathematical interest.
• For ${\displaystyle p}$ the power of some prime number, ${\displaystyle \mathbb {Q} _{p}}$ is still a field. For other composite ${\displaystyle p}$, ${\displaystyle \mathbb {Q} _{p}}$ is a ring, but not a field.
• ${\displaystyle \mathbb {Q} _{p}}$ is not the same as ${\displaystyle \mathbb {R} }$.
• For example, ${\displaystyle {\sqrt {p}}\notin \mathbb {Q} _{p}}$ for any ${\displaystyle p}$, and, for some values of ${\displaystyle p}$, ${\displaystyle {\sqrt {-1}}\in \mathbb {Q} _{p}}$.

#### Hyponyms

• (element of a completion of the rational numbers with respect to a p-adic ultrametric):