Formlære

Formlære, meaning the theory of forms, is a branch of mathematics that studies algebraic forms, primarily homogeneous polynomials in several variables, and their properties under changes of variables. In Danish and Norwegian educational contexts, formlære is a core area within algebra and geometry that provides tools for understanding how expressions behave under linear transformations and scaling.

Key concepts include the degree and homogeneity of a form, representation of numbers by forms, and the classification of forms up to equivalence under linear substitutions. Common examples are quadratic forms in two or more variables, such as ax^2 + bxy + cy^2, and higher-degree forms like cubic or quartic forms. Students learn to manipulate forms, reduce them to canonical or diagonalized representations, and study associated invariants such as discriminants and, for bilinear forms, matrix representations.

Formlære connects to several mathematical disciplines. In geometry, forms describe conic sections and higher-dimensional shapes; in number theory, quadratic forms relate to representations of integers and modular forms; in linear algebra, bilinear and quadratic forms link to matrices, eigenvalues, and orthogonality. The subject frequently serves as a bridge between abstract algebra and geometry, illustrating how algebraic expressions encode geometric and arithmetic structure.

Historically, formlære evolved from invariant theory and the study of quadratic forms in the 19th century and remains a foundational topic in secondary education and introductory algebra courses.