p is an adic number

p -adic number ^[1] is a number-theoretic concept defined for a given fixed prime p as an extension element of the field of rational numbers . This extension is the completion of the field of rational numbers with respect to the p -adic norm , determined on the basis of the properties of divisibility of integers by p .

p -adic numbers were introduced by Kurt Hansel in 1897 ^[2] .

The field of p -adic numbers is usually denoted by $\mathbb {Q} _{p}$ ${\ displaystyle \ mathbb {Q} _ {p}}$ $\ mathbb Q_p$ or $\mathbf {Q} _{p}$ ${\ displaystyle \ mathbf {Q} _ {p}}$ ${\ mathbf Q} _ {p}$ .

Content

Algebraic construction

P -adic integers

Standard Definition

An integer p -adic number for a given prime p is an infinite sequence ^[3] $x=\{x_{1},x_{2},\ldots \}$ ${\ displaystyle x = \ {x_ {1}, x_ {2}, \ ldots \}}$ deductions $x_{n}$ ${\ displaystyle x_ {n}}$ modulo $p^{n}$ ${\ displaystyle p ^ {n}}$ satisfying the condition:

x_{n}\equiv x_{n+1}{\pmod {p^{n}}}.

{\ displaystyle x_ {n} \ equiv x_ {n + 1} {\ pmod {p ^ {n}}}.}

Addition and multiplication of p -adic integers is defined as term-by-term addition and multiplication of such sequences. All axioms of the ring are directly verified for them. The ring of p -adic integers is usually denoted by $\mathbb {Z} _{p}$ ${\ displaystyle \ mathbb {Z} _ {p}}$ .

Definition through the projective limit

In terms of projective limits, the ring of integers $p$ ${\ displaystyle p}$ -adic numbers defined as the limit

\lim _{\leftarrow }\mathbb {Z} /{p^{n}}\mathbb {Z}

{\ displaystyle \ lim _ {\ leftarrow} \ mathbb {Z} / {p ^ {n}} \ mathbb {Z}}

rings $\mathbb {Z} /{p^{n}}\mathbb {Z}$ ${\ displaystyle \ mathbb {Z} / {p ^ {n}} \ mathbb {Z}}$ deductions modulo $p^{n}$ ${\ displaystyle p ^ {n}}$ regarding natural projections $\mathbb {Z} /{p^{n+1}}\mathbb {Z} \to \mathbb {Z} /{p^{n}}\mathbb {Z}$ ${\ displaystyle \ mathbb {Z} / {p ^ {n + 1}} \ mathbb {Z} \ to \ mathbb {Z} / {p ^ {n}} \ mathbb {Z}}$ .

These considerations can be carried out in the case of not only a prime $p$ ${\ displaystyle p}$ , but also of any compound number $m$ ${\ displaystyle m}$ - get the so-called. ring $m$ ${\ displaystyle m}$ -adic numbers, but this ring is unlike $\mathbb {Z} _{p}$ ${\ displaystyle \ mathbb {Z} _ {p}}$ possesses zero divisors ; therefore, the further constructions considered below are not applicable to it.

Properties

Ordinary integers are embedded in $\mathbb {Z} _{p}$ ${\ displaystyle \ mathbb {Z} _ {p}}$ in an obvious way: $x=\{x,x,\ldots \}$ ${\ displaystyle x = \ {x, x, \ ldots \}}$ and are subring.

An example of arithmetic operations on 5-adic numbers.

Taking as the element of the residue class the number $a_{n}=x_{n}\,{\bmod {\,}}{p^{n}}$ ${\ displaystyle a_ {n} = x_ {n} \, {\ bmod {\,}} {p ^ {n}}}$ (in this way, ${\displaystyle 0\leq a_{n}<semantics><mrow class=$ ${\ displaystyle 0 \ leq a_ {n} <p ^ {n}}$ ), we can write each integer p -adic number in the form $x=\{a_{1},a_{2},\ldots \}$ ${\ displaystyle x = \ {a_ {1}, a_ {2}, \ ldots \}}$ unambiguously. This representation is called canonical . Recording each $a_{n}$ ${\ displaystyle a_ {n}}$ in p- number system $a_{n}=b_{n}\ldots b_{2}b_{1}$ ${\ displaystyle a_ {n} = b_ {n} \ ldots b_ {2} b_ {1}}$ and given that $a_{n}\equiv a_{n+1}{\pmod {p^{n}}}$ ${\ displaystyle a_ {n} \ equiv a_ {n + 1} {\ pmod {p ^ {n}}}}$ , it is possible to represent any p -adic number in canonical form in the form $x=\{b_{1},b_{2}b_{1},b_{3}b_{2}b_{1},\ldots \}$ ${\ displaystyle x = \ {b_ {1}, b_ {2} b_ {1}, b_ {3} b_ {2} b_ {1}, \ ldots \}}$ or write as an infinite sequence of digits in a p- number system $x=\{\ldots b_{n}\ldots b_{2}b_{1}\}$ ${\ displaystyle x = \ {\ ldots b_ {n} \ ldots b_ {2} b_ {1} \}}$ . Actions on such sequences are performed according to the usual rules of addition, subtraction and multiplication by a “column” in a p- number system.

In this form of writing, natural numbers and zero correspond to p -adic numbers with a finite number of nonzero digits matching the digits of the original number. Negative numbers correspond to p -adic numbers with an infinite number of nonzero digits, for example, in the quaternary system −1 = ... 4444 = (4).

p -adic numbers

Definition as private fields

p -adic number is an element of the field of quotients $\mathbb {Q} _{p}$ ${\ displaystyle \ mathbb {Q} _ {p}}$ rings $\mathbb {Z} _{p}$ ${\ displaystyle \ mathbb {Z} _ {p}}$ p -adic integers. This field is called the field of p -adic numbers.

Properties

The field of p -adic numbers contains the field of rational numbers .

An example of the division of 5-adic numbers.

It is easy to prove that any p -adic integer not multiple of p is invertible in the ring $\mathbb {Z} _{p}$ ${\ displaystyle \ mathbb {Z} _ {p}}$ , and the multiple p is uniquely written in the form $xp^{n}$ ${\ displaystyle xp ^ {n}}$ , where x is not a multiple of p and therefore reversible, and $n>0$ ${\ displaystyle n> 0}$ . Therefore, any nonzero element of the field $\mathbb {Q} _{p}$ ${\ displaystyle \ mathbb {Q} _ {p}}$ can be written as $xp^{n}$ ${\ displaystyle xp ^ {n}}$ , where x is not a multiple of p , but n is any; if n is negative, then, based on the representation of p -adic integers as a sequence of digits in a p- number system, we can write such a p -adic number as a sequence $x=\{\ldots b_{k}\ldots b_{2}b_{1},b_{0}b_{-1}\ldots b_{n+1}\}$ ${\ displaystyle x = \ {\ ldots b_ {k} \ ldots b_ {2} b_ {1}, b_ {0} b _ {- 1} \ ldots b_ {n + 1} \}}$ , that is, formally represented as a p- fraction with a finite number of digits after the decimal point and, possibly, an infinite number of nonzero digits before the decimal point. The division of such numbers can also be done similarly to the "school" rule, but starting with the lower, and not the highest digits.

Metric Build

Any rational number $r$ ${\ displaystyle r}$ can be imagined as $r=p^{n}{\frac {a}{b}}$ ${\ displaystyle r = p ^ {n} {\ frac {a} {b}}}$ Where $a$ ${\ displaystyle a}$ and $b$ ${\ displaystyle b}$ integers not divisible by $p$ ${\ displaystyle p}$ , but $n$ ${\ displaystyle n}$ - the whole. Then $|r|_{p}$ ${\ displaystyle | r | _ {p}}$ - $p$ ${\ displaystyle p}$ adic norm $r$ ${\ displaystyle r}$ - defined as $p^{-n}$ ${\ displaystyle p ^ {- n}}$ . If a $r=0$ ${\ displaystyle r = 0}$ then $|r|_{p}=0$ ${\ displaystyle | r | _ {p} = 0}$ .

Field $p$ ${\ displaystyle p}$ -adic numbers is the completion of the field of rational numbers with a metric $d_{p}$ ${\ displaystyle d_ {p}}$ defined $p$ ${\ displaystyle p}$ -adic norm: $d_{p}(x,y)=|x-y|_{p}$ ${\ displaystyle d_ {p} (x, y) = | xy | _ {p}}$ . This construction is similar to the construction of the field of real numbers as the completion of the field of rational numbers with the help of the norm, which is the usual absolute value .

Norm $|r|_{p}$ ${\ displaystyle | r | _ {p}}$ continues in continuity to normal at $\mathbb {Q} _{p}$ ${\ displaystyle \ mathbb {Q} _ {p}}$ .

Properties

Each element x of the field of p -adic numbers can be represented as a convergent series

x=\sum _{i=n_{0}}^{\infty }a_{i}p^{i}

{\ displaystyle x = \ sum _ {i = n_ {0}} ^ {\ infty} a_ {i} p ^ {i}}

Where

n_{0}

{\ displaystyle n_ {0}}

Is some integer, and

a_{i}

{\ displaystyle a_ {i}}

- non-negative integers not exceeding

p-1

{\ displaystyle p-1}

. Namely, as

a_{i}

{\ displaystyle a_ {i}}

here are the numbers from the entry x in the number system with the base p . This amount always converges in the metric.

d_{p}

{\ displaystyle d_ {p}}

to myself

x

{\ displaystyle x}

.

p -adic norm $|x|_{p}$ ${\ displaystyle | x | _ {p}}$ satisfies the strong triangle inequality

|x-z|_{p}\leq \max\{|x-y|_{p},|y-z|_{p}\}.

{\ displaystyle | xz | _ {p} \ leq \ max \ {| xy | _ {p}, | yz | _ {p} \}.}

The numbers $x\in \mathbb {Q} _{p}$ ${\ displaystyle x \ in \ mathbb {Q} _ {p}}$ with the condition $|x|_{p}\leq 1$ ${\ displaystyle | x | _ {p} \ leq 1}$ form a ring $\mathbb {Z} _{p}$ ${\ displaystyle \ mathbb {Z} _ {p}}$ p -adic integers, which is the completion of the ring of integers $\mathbb {Z} \subset \mathbb {Q}$ ${\ displaystyle \ mathbb {Z} \ subset \ mathbb {Q}}$ fine $|x|_{p}$ ${\ displaystyle | x | _ {p}}$ .
The numbers $x\in \mathbb {Q} _{p}$ ${\ displaystyle x \ in \ mathbb {Q} _ {p}}$ with the condition $|x|_{p}=1$ ${\ displaystyle | x | _ {p} = 1}$ form a multiplicative group and are called p- adic units.
Collection of numbers $x\in \mathbb {Q} _{p}$ ${\ displaystyle x \ in \ mathbb {Q} _ {p}}$ with the condition $|x|_{p}<1$ ${\ displaystyle | x | _ {p} <1}$ is the main ideal in $\mathbb {Z} _{p}$ ${\ displaystyle \ mathbb {Z} _ {p}}$ with the generating element p .

Metric space $(\mathbb {Z} _{p},d_{p})$ ${\ displaystyle (\ mathbb {Z} _ {p}, d_ {p})}$ homeomorphic to the Cantor set , and the space $(\mathbb {Q} _{p},d_{p})$ ${\ displaystyle (\ mathbb {Q} _ {p}, d_ {p})}$ homeomorphic to the Cantor set with cut point.
For different p norms $|x|_{p}$ ${\ displaystyle | x | _ {p}}$ independent and fields $\mathbb {Q} _{p}$ ${\ displaystyle \ mathbb {Q} _ {p}}$ non-isomorphic.
For any items $r_{\infty }$ ${\ displaystyle r _ {\ infty}}$ , $r_{2}$ ${\ displaystyle r_ {2}}$ , $r_{3}$ ${\ displaystyle r_ {3}}$ , $r_{5}$ ${\ displaystyle r_ {5}}$ , $r_{7}$ ${\ displaystyle r_ {7}}$ , ... such that $r_{\infty }\in \mathbb {R}$ ${\ displaystyle r _ {\ infty} \ in \ mathbb {R}}$ and $r_{p}\in \mathbb {Q} _{p}$ ${\ displaystyle r_ {p} \ in \ mathbb {Q} _ {p}}$ , one can find a sequence of rational numbers $x_{n}$ ${\ displaystyle x_ {n}}$ such that for any p $|x_{i}-r_{p}|_{p}\to 0$ ${\ displaystyle | x_ {i} -r_ {p} | _ {p} \ to 0}$ and $|x_{i}-r_{\infty }|\to 0$ ${\ displaystyle | x_ {i} -r _ {\ infty} | \ to 0}$ .

Applications

If a $F(x_{1},x_{2},\ldots ,x_{n})$ ${\ displaystyle F (x_ {1}, x_ {2}, \ ldots, x_ {n})}$ Is a polynomial with integer coefficients, then solvability for all k comparisons

F(x_{1},x_{2},\cdots ,x_{n})\equiv 0{\pmod {p^{k}}}

{\ displaystyle F (x_ {1}, x_ {2}, \ cdots, x_ {n}) \ equiv 0 {\ pmod {p ^ {k}}}}

equivalent to the solvability of the equation

F(x_{1},x_{2},\cdots ,x_{n})=0

{\ displaystyle F (x_ {1}, x_ {2}, \ cdots, x_ {n}) = 0}

in integer p -adic numbers. A necessary condition for the solvability of this equation in integer or rational numbers is its solvability in rings or, respectively, fields of p -adic numbers for all p , as well as in the field of real numbers. For some classes of polynomials (for example, for quadratic forms) this condition is also sufficient.

In practice, to check the solvability of the equation in p -adic integers, it suffices to check the solvability of this comparison for a certain finite number of values of k . For example, according to Hansel's lemma , for

n=1

{\ displaystyle n = 1}

a sufficient condition for the decidability of the comparison for all natural k is the existence of a simple solution for the comparison modulo p (that is, a simple root for the corresponding equation in the residue field modulo p ). In other words, when

n=1

{\ displaystyle n = 1}

to check for the presence of a root in an equation in p -adic integers, as a rule, it is sufficient to solve the corresponding comparison for

k=1

{\ displaystyle k = 1}

.

p-adic numbers are widely used in theoretical physics ^[4] . The known p-adic generalized functions ^[5] , p-adic analogue of the differentiation operator (Vladimirov operator) ^[6] , p-adic quantum mechanics ^[7] ^[8] , p-adic spectral theory ^[9] , p-adic string theory ^[10] ^[11]

Notes

↑ Pronounced: pe-adic ; respectively: two-adic , triadic , etc.
↑ Kurt Hensel. Über eine neue Begründung der Theorie der algebraischen Zahlen // Jahresbericht der Deutschen Mathematiker-Vereinigung . - 1897. - T. 6 , No. 3 . - S. 83-88 . (German)
↑ Borevich Z. I., Shafarevich I. R. Number Theory, 1985 , p. 25-28 ..
↑ Vladimiriv VS , Volovich IV, Zelenov EI P-adic analysis and mathematical physics // Singapure: World Sci., 1993
↑ V. Vladimirov, “Generalized functions over the field of p-adic numbers” // UMN , 1988, v. 43 (5), p. 17-53
↑ V. Vladimirov, “ On the spectral properties of p-adic pseudodifferential operators of Schrödinger type,” Izv. RAS, Ser. Mat., 1992, v. 56, p. 770–789
↑ Vladimiriv VS , Volovich IV P-adic quantum mechanics // Commun. Math. Phys., 1989, vol. 123, P. 659–676
↑ Vladimiriv VS , Volovich IV P-adic Schrodinger-type equation // Lett. Math. Phys., 1989, vol. 18, P. 43-53
↑ Vladimirov V.S. , Volovich I.V., Zelenov E.I. Spectral theory in p-adic quantum mechanics and representation theory // Izv. USSR Academy of Sciences, vol. 54 (2), p. 275-302, (1990)
↑ Volovich IV P-adic string // Class. Quant. Grav., 1987, vol. 4, P. L83-L84
↑ Frampton PH Retrospective on p-adic string theory // Proceedings of the V. A. Steklov Institute of Mathematics. Collection, No. 203 - M .: Nauka, 1994. - isbn 5-02-007023-8 - S. 287-291.

Literature

Borevich Z. I., Shafarevich I. R. Theory of numbers. - M .: Science, 1985.
Koblitz N. p-adic numbers, p-adic analysis and zeta functions, - M .: Mir, 1982.
Serre J.-P. The course of arithmetic, - M .: Mir, 1972.
Becker B., Vostokov S., Ionin Yu. 2-adic numbers // Quantum . - 1979. - No. 2 . - S. 26-31 .
Konrad K. Introduction to p-adic numbers Summer School "Contemporary Mathematics", 2014 Dubna

[1] Pronounced: pe-adic ; respectively: two-adic , triadic , etc.

[2] Kurt Hensel. Über eine neue Begründung der Theorie der algebraischen Zahlen // Jahresbericht der Deutschen Mathematiker-Vereinigung . - 1897. - T. 6 , No. 3 . - S. 83-88 . (German)

[_221bf300935b26f6-3] Borevich Z. I., Shafarevich I. R. Number Theory, 1985 , p. 25-28 ..

[4] Vladimiriv VS , Volovich IV, Zelenov EI P-adic analysis and mathematical physics // Singapure: World Sci., 1993

[5] V. Vladimirov, “Generalized functions over the field of p-adic numbers” // UMN , 1988, v. 43 (5), p. 17-53

[6] V. Vladimirov, “ On the spectral properties of p-adic pseudodifferential operators of Schrödinger type,” Izv. RAS, Ser. Mat., 1992, v. 56, p. 770–789

[7] Vladimiriv VS , Volovich IV P-adic quantum mechanics // Commun. Math. Phys., 1989, vol. 123, P. 659–676

[8] Vladimiriv VS , Volovich IV P-adic Schrodinger-type equation // Lett. Math. Phys., 1989, vol. 18, P. 43-53

[9] Vladimirov V.S. , Volovich I.V., Zelenov E.I. Spectral theory in p-adic quantum mechanics and representation theory // Izv. USSR Academy of Sciences, vol. 54 (2), p. 275-302, (1990)

[10] Volovich IV P-adic string // Class. Quant. Grav., 1987, vol. 4, P. L83-L84

[11] Frampton PH Retrospective on p-adic string theory // Proceedings of the V. A. Steklov Institute of Mathematics. Collection, No. 203 - M .: Nauka, 1994. - isbn 5-02-007023-8 - S. 287-291.

p is an adic number

Content

Algebraic construction

P -adic integers

Standard Definition

Definition through the projective limit

Properties

p -adic numbers

Definition as private fields

Properties

Metric Build

Properties

Applications

See also

Notes

Literature

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