nichttrivial

Nichttrivial is a German adjective meaning "not trivial." In technical usage it labels objects, results, or arguments that are not straightforward, obvious, or reducible to the given assumptions. The term is widely used in mathematics, logic, and computer science to distinguish substantive content from cases that are considered trivial or self-evident.

Derivation: From nicht meaning "not" and trivial borrowed from Latin trivialis via French and English; compounds with nichttrivial appear in formal writing. Some authors hyphenate as nicht-trivial; others write solid compounds.

In mathematics and related fields, a nontrivial solution means one that is not the obvious or zero solution. Example: for a homogeneous system Ax = 0, the zero vector is trivial; any nonzero vector in the kernel is a nontrivial solution. In topology, a nontrivial topology is one that is not the indiscrete (only empty and whole space) topology; in group theory, a nontrivial group has more than one element.

In other domains, "nichttrivial" describes arguments, constructions, or algorithms that go beyond the simplest cases. A nichttriviale Beweis (nontrivial proof) is a proof that cannot be obtained by a straightforward or direct check of definitions and requires deeper reasoning.

See also: trivial, nontrivial, nontriviality. The English equivalent is "nontrivial," used similarly across mathematics and logic; in German-language texts, nichttrivial may appear as a direct loan or with hyphenation.