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The Jacobi symbol, with its zero value and empty-product convention
Definition
Let and let be an odd positive integer. For odd with canonical prime factorisation , define .
This is the Jacobi symbol of modulo . The prime factors are distinct, every exponent is positive, and each factor on the right is a Legendre symbol (The Legendre symbol, including its zero value). When , the factor list is empty and the finite-product convention (The product of a finite list in a monoid, by recursion, with the empty product () equal to the identity) gives
The value is exactly when , and . Independence from the ordering of the canonical factors, dependence only on , and the stated zero criterion are proved in The Jacobi symbol is well defined on numerator residue classes ↗.
Depends on
- The Legendre symbol, including its zero value
- For $n \ge 1$ and any injective list $p : r \to \mathbb{Z}$ of primes containing every prime divisor of $n$, one has $n = \prod_{i<r} p_i^{\,v_{p_i}(n)}$; the exponents are determined by $n$, and $v_q(n) = 0$ for every prime $q$ outside the list
- The product $g_0 g_1 \cdots g_{n-1}$ of a finite list in a monoid, by recursion, with the empty product ($n = 0$) equal to the identity
- Common divisor, and the greatest common divisor $\gcd(a,b)$, with the convention $\gcd(0,0) := 0$
Used by
- Jacobi symbol one does not imply a square: (2/15)=1 Counterexample
- The Jacobi symbol is well defined on numerator residue classes Proposition
- Quadratic reciprocity for coprime odd Jacobi denominators Theorem
- The Euclidean algorithm computes the Jacobi symbol without factoring the denominator Theorem
- The Jacobi symbol is multiplicative in numerator and denominator Theorem
- The kernel of the Jacobi map and the subgroup of unit squares Theorem
- The two supplementary laws for the Jacobi symbol Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 84 results over 27 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- P. Hackman, Elementary Number Theory, §D.II (standard reference, not scraped)
- A. Gorodnik, Number Theory, Lecture 10, §1 (standard reference, not scraped)
- V. Shoup, A Computational Introduction to Number Theory and Algebra, 2nd ed., §12.2 (standard reference, not scraped)