Give an example of a commutative for each part a) ideal which is not prime b) an
ID: 1941815 • Letter: G
Question
Give an example of a commutative for each parta) ideal which is not prime
b) an ideal which is not maximal
c) an ideal which is maximal
d) a ring with characteristic 0
E) a ring wich characteristic 8
Explanation / Answer
a) If R denotes the ring C[X, Y] of polynomials in two variables with complex coefficients, then the ideal generated by the polynomial Y2 - X3 - X - 1 is a prime ideal. b) In the ring Z of integers the non maximal ideals are the principal ideals generated by a non-prime number. More generally, all non zero prime ideals are non maximal in a principal ideal domain. c) In the ring Z of integers the maximal ideals are the principal ideals generated by a prime number. More generally, all non zero prime ideals are maximal in a principal ideal domain. d) the characteristic is 0. The finite field GF(pn) has characteristic p. There exist infinite fields of prime characteristic. For example, the field of all rational functions over Z/pZ is one such. The algebraic closure of Z/pZ is another example. e) First we consider fields where the size is prime, i.e., n = 1. Such a field is also called a Prime field. An example of such a finite field is the ring Z/pZ. It is a finite field with p elements, usually labelled 0, 1, 2, ..., p-1, where arithmetic is performed modulo p. It is also sometimes denoted Zp, but within some areas of mathematics, particularly number theory, this may cause confusion because the same notation Zp is used for the ring of p-adic integers. Next we consider fields where the size is not prime, but is a prime power, i.e., n > 1. Two isomorphic constructions of the field with 4 elements are (Z/2Z)[T]/(T2+T+1) and Z[f]/(2Z[f]), where f = . A field with 8 elements is (Z/2Z)[T]/(T3+T+1). Two isomorphic constructions of the field with 9 elements are (Z/3Z)[T]/(T2+1) and Z[i]/(3Z[i]). Even though all fields of size p are isomorphic to Z/pZ, for n = 2 the ring Z/pnZ (the ring of integers modulo pn) is not a field. The element p (mod pn) is nonzero and has no multiplicative inverse. By comparison with the ring Z/4Z of size 4, the underlying additive group of the field (Z/2Z)[T]/(T2+T+1) of size 4 is not cyclic but rather is isomorphic to the Klein four-group, (Z/2Z)2. A prime power field with n=2 is also called a binary field. Finally, we consider fields where the size is not a prime power. As it turns out, none exist. For example, there is no field with 6 elements, because 6 is not a prime power. Each and every pair of operations on a set of 6 elements fails to satisfy the mathematical definition of a field.
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.