Galois theory

English

Etymology

Standardly constructed calque of French théorie de Galois (which was used as a section heading in 1870, Camille Jordan, Traité des substitutions et des équations algébriques). First appeared in print in 1893, Bulletin of the New York Mathematical Society.

Ultimately, named after French mathematician Evariste Galois (1811-1832), who first developed the theory to explore how the roots of a given polynomial equation relate to each other.

Pronunciation

• IPA^(key): /ˈɡælwɑ ˈθi.əɹi/

Noun

Galois theory (usually uncountable, plural Galois theories)

1. (algebra, field theory) The branch of mathematics dealing with Galois groups, Galois fields, and polynomial equations. It provides a link between field theory and group theory: it permits certain problems in the former to be reduced to the latter, which in some respects is simpler and better understood.
   • 1989, G. Karpilovsky, Topics in Field Theory, North-Holland, page 299:

     In this chapter we present the Galois theory which may be described as the analysis of field extensions by means of automorphism groups.

   • 1989, Donald S. Passman, Infinite Crossed Products, Academic Press, page 297:

     The Galois theory of noncommutative rings is a natural outgrowth of the Galois theory of fields.

   • 1992, Journal of Contemporary Mathematical Analysis, Volume 27, Allerton Press, page 4,

     Though often our results are prompted by the classical or parallel Galois theories, their proofs are completely different and are based on the set-theoretical approach.

   • 2003, “Introduction”, in Leila Schneps, editor, Galois Groups and Fundamental Groups, Cambridge University Press, page ix:

     Classical Galois theory has developed a number of extensions and ramifications into more specific theories, which combine it with other areas of mathematics or restate its main problems in different situations. Three of the most important are geometric Galois theory, differential Galois theory, and Lie Galois theory, all of which have undergone very rapid development in recent years.

Translations

branch of mathematics

• French: théorie de Galois (fr) f
• Italian: teoria di Galois f

Further reading

• Galois group on Wikipedia.Wikipedia
• Galois field on Wikipedia.Wikipedia
• Differential Galois theory on Wikipedia.Wikipedia
• Grothendieck's Galois theory on Wikipedia.Wikipedia
• Galois connection on Wikipedia.Wikipedia
• Galois theory on Encyclopedia of Mathematics
• Galois theory of rings on Encyclopedia of Mathematics
• Galois Theory on Wolfram MathWorld