HW6_322_F09

# HW6_322_F09 - Fall 2009 MAT 322 ALGEBRA WITH GALOIS THEORY...

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Fall 2009 MAT 322: ALGEBRA WITH GALOIS THEORY PROBLEM SET #6 Week #6: October 26th - November 1st, and Week #7: November 9th - November 15th. Topics : Euler’s totient, characteristic of a ﬁeld, prime and irreducible elements, GCD’s and LCM’s, Euclidean PID UFD, F [ x ] and Z [ ω ] are Euclidean, Euclid’s algorithm, content of a polynomial, the Gauss lemma, R [ x ] is a UFD when R is a UFD, ﬁeld extensions, the prime subﬁeld, multiplicativity of the degree, the minimal polynomial Irr( α, F, x ), cyclotomic polynomials Φ n ( x ). Read : FGT [7-10] (again, check out Lang’s Algebra , Ch. II and Ch. V) Problems due Friday, November 13th, at 4:00 pm in Martin Luu’s mailbox: Problem 1 : Let E be a ﬁnite ﬁeld extension of F , and let R be a subring of E containing F . Show that R is necessarily a subﬁeld of E . Problem 2 : Compute the residue of 117 117 modulo 17 . Problem 3

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## This note was uploaded on 05/07/2010 for the course MAT 322 at Princeton.

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HW6_322_F09 - Fall 2009 MAT 322 ALGEBRA WITH GALOIS THEORY...

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