Évariste Galois in the context of Galois group

In mathematics, a Galois module is a G-module, with G being the Galois group (named for Évariste Galois) of some extension of fields. The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory.

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that has a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are the integers mod ${\displaystyle p}$ when ${\displaystyle p}$ is a prime number.

The order of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number ${\displaystyle p}$ and every positive integer ${\displaystyle k}$ there are fields of order ${\displaystyle p^{k}}$. All finite fields of a given order are isomorphic.

Arthur Cayley FRS (/ˈkeɪli/; 16 August 1821 – 26 January 1895) was an English mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics, and was a professor at Trinity College, Cambridge for 35 years.

He postulated what is now known as the Cayley–Hamilton theorem—that every square matrix is a root of its own characteristic polynomial, and verified it for matrices of order 2 and 3. He was the first to define the concept of an abstract group, a set with a binary operation satisfying certain laws, as opposed to Évariste Galois' concept of permutation groups. In group theory, Cayley tables, Cayley graphs, and Cayley's theorem are named in his honour, as well as Cayley's formula in combinatorics.

Auguste Bravais (French pronunciation: [oɡyst bʁavɛ]; 23 August 1811, Annonay, Ardèche – 30 March 1863, Le Chesnay, France) was a French physicist known for his work in crystallography, the conception of Bravais lattices, and the formulation of Bravais law. Bravais also studied magnetism, the northern lights, meteorology, geobotany, phyllotaxis, astronomy, statistics and hydrography.

He studied at the Collège Stanislas in Paris before joining the École Polytechnique in 1829, where he was a classmate of groundbreaking mathematician Évariste Galois, whom Bravais actually beat in a scholastic mathematics competition. Towards the end of his studies he became a naval officer, and sailed on the Finistere in 1832 as well as the Loiret afterwards. He took part in hydrographic work along the Algerian Coast. He participated in the Recherche expedition and helped the Lilloise in Spitzbergen and Lapland.