View source for -adic

{{also|adic}}
==English==

===Etymology===
{{back-formation|en|monadic}}, etc., from {{der|en|grc|-άς}} (genitive {{m|grc|-άδος}}) + {{m|grc|-ικός}} (English {{m|en|-ad}} + {{m|en|-ic}}). Compare related {{m|en|adicity}} and Latinate {{m|en|-ary}}.

The algebraic sense is by analogy with {{back-formation|en|notext=1|p-adic}}, since <math>\mathbb{Z}_{(p)}</math> equipped with the <math>(p)</math>-adic topology passes to the ring of {{w|p-adic numbers}} under the operation of [[complete|completion]].

===Pronunciation===
* {{audio|en|LL-Q1860 (eng)-Flame, not lame-&#45;adic.wav|a=US}}

===Suffix===
{{head|en|suffix|cat2=adjective-forming suffixes}}

# {{lb|en|mathematics|computing}} Having a specified [[adicity]].
# {{lb|en|algebra|of a [[topology]] on a [[module]] <math>M</math> over a [[ring]] <math>R</math>|combined with a symbol (here <math>I</math> but also often <math>p</math>, <math>m</math>, or <math>a</math>) representing an [[ideal]] of <math>R</math>}} Such that the sets <math>x+I^nM</math> (where <math>x \in M</math>, <math>n</math> a non-negative integer) form a [[basis]] for the topology.

====Usage notes====
Combined with prefixes derived (usually) from Greek names for numbers to make adjectives meaning "having a certain number of [[argument]]s" (said of [[function]]s, [[relation]]s, etc, in mathematics and functions, [[operator]]s, etc, in computing).

Since there is a unique <math>I</math>-adic topology for any given <math>I</math>, one often speaks of ''the'' <math>I</math>-adic topology on <math>M</math>.

====Synonyms====
* {{sense|adicity}} {{l|en|-ary}}<!-- Latin -->

====Derived terms====
{{suffixsee|en}}

====Related terms====
{{col|en
|decadic
|duadic
|dyadic
|enneadic
|heptadic
|hexadic
|monadic
|[[n-adic|''n''-adic]]
|niladic
|nonadic
|octadic
|p-adic
|pentadic
|polyadic
|tetradic
|triadic
|variadic
}}

====Translations====
{{trans-top|having a number of arguments}}
* Finnish: {{t|fi|-adinen}}
* French: {{t|fr|-adique}}
* German: {{t|de|-adisch}}
* Hungarian: {{t|hu|-adikus}}
* Italian: {{t|it|-adico}}
* Polish: {{not used|pl}}
* Romanian: {{t|ro|-adic}}
{{trans-bottom}}

===See also===
* [[Appendix:English arities and adicities]]

===Anagrams===
* {{anagrams|en|a=acdi|ACID|Daic|acid|cadi|caid}}