in the century from 1832 to noether's death in 1935, the field of mathematics – specifically algebra – underwent a profound revolution, whose reverberations are still being felt. mathematicians of previous centuries had worked on practical methods for solving specific types of equations, e.g., cubic, quartic, and quintic equations, as well as on the related problem of constructing regular polygons using compass and straightedge. beginning with carl friedrich gauss's 1832 proof that prime numbers such as five can be factored in gaussian integers, évariste galois's introduction of permutation groups in 1832 (although, because of his death, his papers were published only in 1846, by liouville), william rowan hamilton's discovery of quaternions in 1843, and arthur cayley's more modern definition of groups in 1854, research turned to determining the properties of ever-more-abstract systems defined by ever-more-universal rules. noether's most important contributions to mathematics were to the development of this new field, abstract algebra.