3adic

3adic typically refers to the 3-adic numbers, the p-adic numbers associated with the prime p = 3. They form a field, denoted Q3, obtained by completing the rational numbers with respect to the 3-adic absolute value |·|3. This valuation is defined by v3(x), the exponent of 3 in x’s prime factorization (with v3(0) = ∞), and |x|3 = 3−v3(x). Concretely, Q3 consists of limits of Cauchy sequences of rationals under the metric d(x,y) = |x − y|3.

The subring of elements with |x|3 ≤ 1 is the 3-adic integers Z3. Every element of Z3 can be written uniquely as a convergent series x = ∑i≥0 ai3i with digits ai ∈ {0,1,2}. General elements of Q3 can be represented as x = ∑i≥v ai3i for some integer v and digits ai ∈ {0,1,2} with av ≠ 0, allowing finitely many negative powers.

The 3-adic norm makes Q3 a complete, non-Archimedean (ultrametric) field. This implies strong forms of the triangle inequality, such as |x + y|3 ≤ max(|x|3, |y|3). The 3-adic integers Z3 form a compact ring and serve as a natural setting for local studies in number theory.

Applications and significance include local fields in algebraic number theory, Hensel’s lemma for lifting solutions from modulo 3 to higher powers, and various areas of p-adic analysis and arithmetic geometry. The 3-adic world provides a complementary perspective to real and complex analysis in the study of Diophantine problems and modular forms.