Galois theory, originally introduced by Evariste Galois, provides a connection between field theory and group theory. Its fundamental theorem allows certain problems in field theory to be reduced to problems in group theory, where they become simpler and easier to understand. The theory grew out of the classical problem of solving polynomial equations by radicals: Paolo Ruffini and Niels Henrik Abel had shown that the general quintic admits no such solution, while Galois went further and gave a precise criterion for deciding whether any particular quintic or higher degree polynomial is solvable by radicals.
Facts
Central QuestionGalois theory asks which polynomial equations can be solved by a formula using only radicals, and shows that the answer is controlled by the group of permutations of the polynomial's roots. 1 Key DebateThe inverse Galois problem, whether every finite group occurs as the Galois group of some field extension of the rational numbers, remains open; only partial results are known, such as Igor Shafarevich's proof that every solvable finite group occurs this way and the confirmed existence of solutions for all 26 sporadic simple groups. 1 Classification
Pure or Applied Connections
Sources
1. Galois Theory (Wikipedia)
Wikimedia FoundationLead section
Galois introduced the subject for studying roots of polynomials.
Inverse Galois problem section
On the other hand, it is an open problem whether every finite group is the Galois group of a field extension of the field Q of the rational numbers. Igor Shafarevich proved that every solvable finite group is the Galois group of some extension of Q.
History section, Ruffini and Abel passage
While Ruffini and Abel established that the general quintic could not be solved, some particular quintics can be solved.
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