Abstr Alg, 26B: Division Algorithm & Corollaries, Irreducible Polynomials over Field, Maximal Ideals

Abstract Algebra, Lecture 26B, Division Algorithm and Corollaries, Irreducible Polynomials over a Field and Maximal Ideals.

(0:00) Meaning of the long division from the end of lecture A in terms of the division algorithm equation.
(1:30) Definition of what it means for one polynomial to divide another in D[x], where D is an integral domain.
(2:36) Definition of a zero (or root) and a zero of multiplicity k.
(5:22) Remainder Theorem (relate to the example), Factor Theorem, and a theorem giving an upper bound of the number of zeros of a polynomial over a field, counting multiplicity (the upper bound is the degree of the polynomial).
(9:01) Ideals in a ring of polynomials over a field are generated by nonzero polynomials of minimum degree in that ideal (and a reminder of what a principal ideal is and note that F[x] is a principal ideal domain (PID)...an analog in ring theory of a cyclic group in group theory).
(14:28) Irreducibility/reducibility of a polynomial over a field (generalizes the concept of a prime number).
(16:21) Reducibility/Irreducibility Test for a degree 2 or 3 polynomial over a field.
(16:51) Consider whether f(x) = x^2 + 1 is reducible or irreducible over Z2 and Z3 (the answers are different).
(20:43) Relate the results to factor rings and maximal ideals (a principal ideal generated by p(x) in F[x] is maximal iff p(x) is irreducible over F iff the factor ring F[x]/A is a field, where A is the principal ideal generated by p(x)).
(22:01) Calculation of the multiplicative inverse of x + A in the factor ring (quotient ring) Z3[x]/A, where A is the principal ideal generated by x^2 + 1.

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[ldb]

(x+1)^2 = x^2 + 1 - "you were right in Z2 but wrong in R" at 19:30 didn't seem appreciated by the students, but I appreciate it. I have math teaching experience so that was hilarious! Should've brought the house down.

[sherryj]

Hello Sir, At 12:45 you are saying this is because of this theorem that we have to take degree of zero polynomial as not defined.But, this theorem, talks about only those g(x) which are non zero .So, we will be simply neglecting the degree of zero polynomial while applying this theorem. How does the degree of zero polynomial affects this theorem in any way?
thanks

[sherryj]

Sorry to say sir, but last few minutes of lecture were not so good. Apologize, but this is what i felt.