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bigfooted
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I want to get a decent introduction into group theory and Galois groups. Can somebody recommend a good book that I can use for self-study? The book of Stewart - Galois Theory looks promising.
A Galois group is a mathematical concept named after the French mathematician Évariste Galois. It is a group of automorphisms that preserve the algebraic structure of a field extension. In simpler terms, it is a group of symmetries that allow us to understand the roots of a polynomial equation.
Understanding Galois groups is important because it allows us to solve polynomial equations and study the properties of fields. It also has applications in other fields such as cryptography and coding theory.
Some examples of Galois groups include the permutation group Sn for degree n, the cyclic group Cn for degree n, and the dihedral group Dn for degree n. These groups have different structures and properties, and studying them can provide insights into the behavior of polynomial equations.
There are many resources available for learning about Galois groups, including textbooks, online courses, and lectures. It is recommended to have a solid foundation in abstract algebra before diving into the study of Galois groups.
Yes, there are various practical applications of Galois groups in fields such as coding theory, cryptography, and number theory. For example, Galois groups are used in error-correcting codes in telecommunication systems and in encryption algorithms for secure communication.