The Langlands program in mathematics relates to the attempt to understand this as applied to the algebraic closure of the rational numbers. For the polynomial x to the fourth plus three x squared plus five, it associates the dihedral group of order 8. It involves eponymous groups of automorphisms of a field that preserve a given subfield. The fundamental theorem of this subject establishes a one-to-one correspondence between subgroups of its namesake group and intermediate fields of a field extension. Although its most 
The Langlands program in mathematics relates to the attempt to understand this as applied to the algebraic closure of the rational numbers. For the polynomial x to the fourth plus three x squared plus five, it associates the dihedral group of order 8. It involves eponymous groups of automorphisms of a field that preserve a given subfield. The fundamental theorem of this subject establishes a one-to-one correspondence between subgroups of its namesake group and intermediate fields of a field extension. Although its most famous result was proved earlier by Abel, the establishment of this field led to a deeper understanding of the insolubility of the quintic equation by radicals. For 10 points, what is this branch of mathematics named for a French mathematician who was killed in a duel at age 20?
The Langlands program in mathematics relates to the attempt to understand this as applied to the algebraic closure of the rational numbers.
Galois theory