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Abstract Alg, 35A: Spitting Field for Irreducible Cubic Polynomial x^3+x^2+1 over Finite Field Z2
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Abstract Algebra, Lecture 35A. Field Theory and Galois Theory, Part 6.
(0:00) Plan for upcoming lectures and the current lecture.
(1:44) Story from when I was in high school.
(5:16) Main Example: Find a splitting field for f(x) = x^3 + x^2 + 1 over F = Z2.
(6:53) Description of the factor (quotient) ring E = Z2[x]/A, where A is the ideal generated by f(x).
(7:52) Does f(x) have a zero (root) in the field extension E? Does is split in E? Is E a splitting field for f(x) over F?
(10:04) It definitely has a zero in E, the coset x + A. Confirm with coset calculation.
(13:09) Write E without coset notation, doing operations modulo 2 and modulo f(x) = x^3 + x^2 + 1 (so x^3 = -x^2 - 1 = x^2 + 1).
(19:05) If beta = x, another zero of f(x) is beta^2.
(22:00) Find the third zero of f(x) in E with synthetic division.
(25:26) Write f(x) as a product of linear factors.
(27:19) So f(x) does split in E. And is the splitting field of f(x) over F. Use Mathematica to show addition and multiplication table for E. Comments about the isomorphism classes of E under addition and the units of E, U(E), under multiplication to argue why no proper subfield of E is a splitting field for f(x) over F.
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