MATH 4E03/6E03, Winter 2019
The quadratic formula for solving polynomials of degree 2 has been known since antiquity. Less familiar are the corresponding formulas for solving polynomials of degrees 3 and 4, discovered by mathematicians in 16th century Renaissance Italy. These expressions are more complicated than their quadratic counterpart, but the fact that such formulas are possible is perhaps not too surprising. It is therefore unexpected that no such formulas exist for solving polynomials of degrees 5 and higher. Why should this be so? Galois theory was invented precisely to address this and related questions about polynomials. In addition to giving us an understanding to the roots of polynomials, Galois theory also gave birth to many of the central concepts of modern algebra, including groups and fields. Moreover, there is the human drama of Evariste Galois, whose untimely death at age 20 left us with the brilliant but not fully developed ideas that eventually led to Galois theory. More than just great history, Galois theory is also great mathematics. It gives us a surprising link between group theory and the roots of polynomials, accomplished in a theory of great elegance in its presentation. Galois theory is often described as one of the most beautiful parts of mathematics.
INSTRUCTOR: A. Nicas
An introduction to Galois Theory. Topics include: field extensions, splitting fields, normality and separability, Galois extensions, finite fields, solvability by radicals, cyclic extensions, cyclotomic extensions, algebraic closure, classical constructions, computations of Galois groups.
Three lectures; one term
Prerequisite(s): MATH 3E03 or 3GR3
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