Solve and answer the following problems as indicated. 1) List down all zero divisors in ℤ_18 in increasing order, separated by commas and without spaces. 2) Create the multiplication table for {0,3,6,9,12} under multiplication modulo 15. This is a subring of ℤ_15. Determine its unity. 3) Find all the roots of f(x)=x^2+3x+2 in ℤ_6. List them as you did in #31. 4) The ideals of ℤ_n are precisely its additive subgroups. Draw the lattice diagram of subgroups
Solve and answer the following problems as indicated.
1) List down all zero divisors in ℤ_18 in increasing order, separated by commas and without spaces.
2) Create the multiplication table for {0,3,6,9,12} under multiplication modulo 15. This is a subring of ℤ_15. Determine its unity.
3) Find all the roots of f(x)=x^2+3x+2 in ℤ_6. List them as you did in #31.
4) The ideals of ℤ_n are precisely its additive subgroups. Draw the lattice diagram of subgroups and COMPLETELY list down the maximal ideals for ℤ_36. If j<k<l<..., list the ideals as (j),(k),(l),... (Do not use the ellipsis since you have to write them all down)
5) Let f(x),g(x) be polynomials in ℤ_5[x]. Find f(x)g(x). Write the polynomial as an infinite sequence (a_0,a_1,...,a_n,...) where a_i is one of the residues 0,1,2,3,4 modulo 5. Do not skip any term in between a_0 and a_n.
f(x) = 4x^3 + 2x^2 + x + 3
g(x) = 3x^4 + 3x^3 + 3x^2 + x + 4
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Solve and answer the following problems as indicated.
2) Create the multiplication table for {0,3,6,9,12} under multiplication modulo 15. This is a subring of ℤ_15. Determine its unity.
3) Find all the roots of f(x)=x^2+3x+2 in ℤ_6. List them in increasing order, separated by commas and without spaces.
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