Abstract Algebra: Group of Units, Integral Domains, Fields, Polynomial Ring Examples & Calculations

Abstract Algebra Ring Theory Unit. Z8 has a multiplication table that shows its units and zero divisors. It group of units U(Z8) is a non-cyclic group of order 4. Zp is both an integral domain and a field if and only if p is prime. Z is an integral domain but not a field. The group of units of the commutative ring with unity Z is U(Z)={1,-1}. The Gaussian integers Z[i] is an integral domain but not a field. The group of units of the ring Z[i] is U(Z[i])={1,-1,i,-i}. The polynomial rings Z[x], R[x], and Z2[x] are integral domains but not fields. When we multiply elements in Z2[x], the coefficients must be reduced with mod 2 arithmetic. Show a long division and synthetic division example: in Z2[x], 1 is a root of multiplicity 3 for the polynomial x^5+x^3+x^2+1. Also, x^2+x+1 is irreducible over the finite field Z2 (it cannot be factored in Z2[x]).

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⏱️TIMESTAMPS⏱️
(0:00) Multiplication table in the commutative ring Z8 (with unity 1).
(1:29) Zero divisors in Z8 are 2, 4, and 6, so Z8 is not an integral domain.
(2:14) Units in Z8 are 1, 3, 5, and 7.
(3:37) U(Z8)=U(8) is not cyclic. It is isomorphic to Z2 + Z2 (the external direct product of the cyclic group Z2 with itself).
(5:13) Group of units U(R) in any commutative ring with unity R.
(7:18) Zp is an integral domain (and field) if and only if p is prime.
(8:28) Every finite integral domain is a field. There are infinite integral domains which are not fields.
(10:38) The ring of integers Z is an integral domain but not a field.
(11:11) U(Z) = {1,-1}, a cyclic group of order 2 (the group of units is isomorphic to the group Z2)
(12:12) The ring of Gaussian integers Z[i] is an integral domain but not a field.
(12:56) U(Z[i])={1,i,-1,-i} is a cyclic group of order 4 (the group of units is isomorphic to the group Z4).
(14:59) The polynomial ring Z[x] is an integral domain but not a field. Also discuss polynomial vector spaces.
(19:10) U(Z[x])={1,-1} is a cyclic group of order 2.
(20:54) The polynomial ring R[x] is an integral domain which is not a field.
(22:05) U(R[x]) = R*, the nonzero constant polynomials (the group of units is isomorphic to the multiplicative group of nonzero real numbers).
(25:46) The polynomial ring Z2[x] is an integral domain which is not a field (it is an infinite ring).
(28:36) U(Z2[x])={1} is a trivial group.
(29:01) For any given degree, the number of polynomials of that degree is finite (even though the ring Z2[x] itself is infinite).
(30:32) Multiplying in Z2[x] (examples of multiplication in Z2[x]).
(33:49) Polynomial factorization discussion.
(34:44) Long division example in Z2[x] (long division mod 2).
(41:58) Synthetic division example in Z2[x] (synthetic division mod 2).
(50:00) Division algorithm expansion using a quotient and remainder when the remainder is nonzero.
(50:53) x^2+x+1 is irreducible over Z2 (it cannot be factored in Z2[x]).
(51:47) The role of factor rings (a.k.a. quotient rings) will be to construct finite fields other than Zp.
(53:18) Definition of an ideal of a ring (a two-sided ideal in a ring), which is analogous to the concept of a normal subgroup of a group).
(55:58) Ideal test (closed under subtraction and "super-closed" under multiplication).
(57:39) Factor ring R/A multiplication is well-defined when the subring A is an ideal of R. The cosets in a factor ring (quotient ring) are written with additive notation.
(58:49) There are subrings which are not ideals.

#ringtheory #integraldomain #polynomialring

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