Difference between revisions of "P-adic values"
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(Statement of Ostrowski's theorem is incorrect: we have to get an absolute value from the p-adic valuation described -- the valuation itself is not an absolute value.) |
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Given a [[prime number]] p the '''p-adic value''' is the function, denoted <math>v_p</math> which takes as its argument a natural number n and returns the power of p appearing in the [[prime factorization]] of that number (equivalently, the highest power of p which divides n): | Given a [[prime number]] p the '''p-adic value''' is the function, denoted <math>v_p</math> which takes as its argument a natural number n and returns the power of p appearing in the [[prime factorization]] of that number (equivalently, the highest power of p which divides n): | ||
| − | <math>v_p(x)=\max\{n:p^n\mid x\}</math>. For example, the p-adic values of 60 for p=2,3,5,7,11,13... are 2,1,1,0,0,0,.... | + | <math>v_p(x)=\max\{n:p^n\mid x\}</math>. For example, the p-adic values of 60 for p=2,3,5,7,11,13... are 2,1,1,0,0,0,.... One can associate with the p-adic valuation an absolute value <math>|n|_p=p^{-v_P(n)}</math>. |
By convention, <math>v_p(0)=\infty</math> for all primes p. | By convention, <math>v_p(0)=\infty</math> for all primes p. | ||
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* The [[Fundamental Theorem of Arithmetic|fundamental theorem of arithmetic]] can be restated compactly using p-adic values: For all natural numbers n, <math>n=\prod_pp^{v_p(n)}</math> where p ranges over all primes. | * The [[Fundamental Theorem of Arithmetic|fundamental theorem of arithmetic]] can be restated compactly using p-adic values: For all natural numbers n, <math>n=\prod_pp^{v_p(n)}</math> where p ranges over all primes. | ||
* p-adic values can be extended to the rational numbers by defining <math>v_p(x/y)=v_p(x)-v_p(y)</math> for all integers x,y. | * p-adic values can be extended to the rational numbers by defining <math>v_p(x/y)=v_p(x)-v_p(y)</math> for all integers x,y. | ||
| − | * Ostrowski's theorem states that the only absolute values on the field of rational numbers are the real [[absolute value]] (which some mathematicians view as the "prime at infinity") and the p-adic values. | + | * Ostrowski's theorem states that the only absolute values on the field of rational numbers are the real [[absolute value]] (which some mathematicians view as the "prime at infinity") and the p-adic absolute values described above. |
p-adic values are used most commonly in [[number theory]] and [[algebra]], especially in the theory of [[commutative]] [[ring]]s. | p-adic values are used most commonly in [[number theory]] and [[algebra]], especially in the theory of [[commutative]] [[ring]]s. | ||
Revision as of 01:01, January 19, 2009
Given a prime number p the p-adic value is the function, denoted Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_p} which takes as its argument a natural number n and returns the power of p appearing in the prime factorization of that number (equivalently, the highest power of p which divides n): . For example, the p-adic values of 60 for p=2,3,5,7,11,13... are 2,1,1,0,0,0,.... One can associate with the p-adic valuation an absolute value .
By convention, for all primes p.
Here are some important properties of p-adic values:
- p-adic values convert multiplication into addition (akin to the logarithm function): Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_p(xy) = v_p(x) + v_p(y)} .
- p-adic values satisfy the archimedean inequality: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_p(x+y) \le \min\{v_p(x),v_p(y)\}} .
- Equality holds in the above so long as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_p(x)\ne v_p(y)} .
- The fundamental theorem of arithmetic can be restated compactly using p-adic values: For all natural numbers n, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=\prod_pp^{v_p(n)}} where p ranges over all primes.
- p-adic values can be extended to the rational numbers by defining Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_p(x/y)=v_p(x)-v_p(y)} for all integers x,y.
- Ostrowski's theorem states that the only absolute values on the field of rational numbers are the real absolute value (which some mathematicians view as the "prime at infinity") and the p-adic absolute values described above.
p-adic values are used most commonly in number theory and algebra, especially in the theory of commutative rings.
Completing the field of rational numbers with respect to p-adic values yiels the field of p-adic numbers.