algebraischen

Algebraischen is the inflected form of the German adjective algebraisch, meaning algebraic. It is used to describe nouns related to algebra or to express properties that are characteristic of algebra.

In German grammar, adjectives are declined to reflect gender, number, and case. The form algebraischen appears in several common contexts. After definite determiners, it often appears with the ending -en in plural or certain singular cases, as in die algebraischen Strukturen (the algebraic structures) or den algebraischen Beweisen (the algebraic proofs). It also occurs in phrases like der algebraischen Gleichung (the algebraic equation, feminine singular in a specific case) or im algebraischen Sinn (in the algebraic sense, masculine dative). In the plural nominative and accusative, as well as in dative plural forms after determiners, algebraischen is a typical ending.

Usage in mathematics and related fields is widespread in German texts. Common collocations include algebraische Gleichungen (algebraic equations), algebraische Strukturen (algebraic structures), algebraische Methoden (algebraic methods), and algebraische Objekte (algebraic objects). The form is also found in more general discussions of algebra as a discipline, not only in technical contexts.

Etymology: the word algebraischen derives from algebra, which in turn comes from the Arabic al-jabr, transmitted through Latin and other European languages. The adjective form algebraisch was created in German to describe qualities, methods, or objects associated with algebra.