In the quotient ring Z2[x]/(z6 + 1), verify that the ideal consisting of all multiples of g(x) = ...
(Abstract Algebra-Ring Theory) Consider the quotient ring Z2[x]/I, where I is the ideal consisting of all (polynomial) multiples of x3 + 1. How many elements are in this quotient ring? Show that the quotient ring is not an integral domain by finding a zero divisor.
{Abstract Algebra - Ring Theory} The ideal, I, contains all multiples of x5+x2 1 in the polynomial ring Z2x]. Assuming that x31- 1 (mod I), explain why the order of x in the quotient ring cannot be smaller than 31.
4. Show that the polynomial g(x) = x++x+1 is irreducible over Z2. In the quotient ring Z2[x]/(g(x)) let S = x+(g(x)), so that Z2[x]/(g(x)) = Z2(). Express 85 and (82 +1)-1 in the form a + b8 + 082 +883, where a, b, c, d e Z2.
1. Verify that the set V, consisting of all scalar multiples of (1,-1, -2) is a subspace of R. 2. Let V, be the set of all 2 x 3 matrices. Verify that V, is a vector space. 3. Let A=(1-11) Let V, be the set of vectors x € R such that Ax = 0. Verify that V, is a subspace of R. Compare V, with V.
Thee part question. Please answer all parts! Let E be a field of characteristic p > 0 (we proved p must always be prime). Verify that the ring homomorphism X : Z → E determined by sending χ : 1-1 E (the unity in E) ( so x(n)-n 1E wheren1E 1E 1E (n-times), x(-n)- nle for any n 1,2,3,... and X(0) 0E by definition of χ) is in fact a ring homomorphism with ker(X) = pZ. Úse the fundamental homomorphism...