p-adic norm
The "p-adic norm" is crucial in number theory and algebra, providing a unique way to measure the size of numbers based on their divisibility by a prime number.
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Definition
C2Number Theory
(technical, academic)A function that assigns a non-negative real number to each rational number, indicating how divisible the number is by a fixed prime number.
Example
- The p-adic norm of 25 with respect to the prime 5 is 1/25, reflecting its high divisibility by 5.
C2Algebra
(technical, academic)A norm on a vector space defined over a field with a discrete valuation, generalizing the concept of p-adic absolute value.
Example
- In algebra, the p-adic norm helps in defining metrics on vector spaces over fields with discrete valuations.