NichttrivialenAutomorphismus

Nichttrivialen is a term used in German mathematical writing to denote a nontrivial object, solution, or property. The English equivalent is nontrivial. In mathematics, something is described as nontrivial when it cannot be reduced to a trivial or obvious case, or when its existence or content is not immediately evident from the definition or context.

In practice, nontriviality signals that a result or construction has substantive content beyond the most obvious instance. For example, a nontrivial solution to an equation is a solution other than the obvious one, such as a nonzero vector in a homogeneous system. In topology or algebra, a nontrivial element or structure is one that is not merely the identity or a degenerate case. In complex analysis and number theory, the term is often used to distinguish meaningful phenomena from those that arise by default from definitions.

Etymology and usage notes: the term combines nicht (not) with trivial (trivial) and is common in German mathematical prose. When German authors translate their work into English, nontrivial is the standard equivalent. Some contexts impose stricter meanings, differentiating between nontriviality in a construction versus nonexistence or nondegeneracy.

See also: trivial, nontrivial, trivial solution, nontrivial zero, nontrivial subgroup.