Boolean Ring Example, Division Algorithm Ex in Zp[x], Corollaries of Division Algorithm over field F

Abstract Algebra: The power set P(S) of S={a,b,c} is a Boolean ring example when addition is defined using the symmetric difference for addition and intersection for multiplication. This ring is commutative with unity. The zero (additive identity) is the empty set while the whose set S is the unity (multiplicative identity). Use long division of polynomials mod 7 to illustrate a Division Algorithm Example for coefficients fro a field (in this case, the field Zp[x] = Z7[x]). Also discuss corollaries of the Division Algorithm (the Remainder Theorem and Factor Theorem). Also introduced are: reducibility and irreducibility of a polynomial over a field (Example: x^2-2 is irreducible over Q but x^2-2 is reducible over R) and principal ideal domains. Field extensions and Galois theory are also briefly previewed.

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⏱️TIMESTAMPS⏱️
(0:00) Boolean ring example (power set of S={a,b,c} is a ring when addition is the symmetric difference of two sets and multiplication is the intersection of two sets).
(2:17) The empty set is the additive identity of P(S).
(3:47) The entire set S={a,b,c} is the unity (multiplicative identity) of P(S).
(4:49) Example calculations (sum, product, power, additive inverse).
(5:46) Every element is idempotent, so P(S) is a Boolean ring.
(7:08) As a group under addition, it is isomorphic to Z2+Z2+Z2.
(8:53) Rings of polynomials and factorization are our main emphasis for the rest of the semester.
(9:37) The idea of a field extension.
(11:02) Galois group of a field extension E over F will be important (automorphisms of E that fix F (fixed field)).
(14:12) In Zp[x]=Z7[x], use polynomial long division (mod 7) to find the quotient q(x) and remainder r(x) when dividing f(x)=5x^4+3x^3+1 by g(x)=3x^2+2x+1 (to illustrate the Division Algorithm).
(22:46) Check by polynomial multiplication mod 7 (in Zp[x]=Z7[x]).
(27:20) Division Algorithm in F[x], where F is a field.
(30:21) Corollary 1: Remainder Theorem (over a field).
(30:56) Idea of proof of the Remainder Theorem.
(34:13) Corollary 2: Factor Theorem (over a field)
(36:01) A polynomial of degree n over a field has at most n zeros, counting multiplicity.
(37:47) Introduction to irreducibility and reducibility of a polynomial over an integral domain or over a field.
(38:42) Irreducibility generalizes prime numbers to rings of polynomials
(40:13) Irreducibility over a field.
(40:50) Example: x^2-2 is irreducible over Q (the field of rational numbers).
(42:22) x^2-2 is reducible over R (the field of real numbers)
(43:41) Principal Ideal Domain definition (PID, or P.I.D.)
(44:38) Z is a PID (the ring of integers forms a principal ideal domain).
(46:08) F[x] is a PID (the polynomial ring with coefficients from a field F forms a principal ideal domain.
(46:58) An ideal I in F[x] is generated by a polynomial g(x) if and only if g(x) is a polynomial of minimum degree in the ideal I.
(49:03) The polynomials with f(0)=0 forms an ideal and is generated by x.

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[imantawfiq]

Hello Mr
Excuse me can you help me for this question ( Let 𝐾={𝑚∈𝑅∶𝑚>0}. if addition and multiplication on 𝐾 defined by: 𝑚+𝑛 = 𝑚𝑛 and 𝑚.𝑛 = 𝑚^ln(𝑛) show that 𝐾 is a commutative unital ring)