Question

Let p(x) = x2 + x + 2 in Z3 [x]. a) List the distinct cosets...

Let p(x) = x2 + x + 2 in Z3 [x].

a) List the distinct cosets of E = Z3 [x] / <x2 + x + 2>

b) Write the addition and multiplication table of E = Z3 [x] / <x2 + x + 2>

c) Identify a subfield of E that is isomorphic to Z3

d) Does p(x) have any zeros in E? (Hint: since E is a field, what is the maximum number of zeros p(x) can have in E?)

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Homework Answers

Answer #1

(a) Distinct co-sets in are of the form ..

(b) .

with these rules we can find out multiplication and addition table.

(c) Note that is identity in . Because for any , we have . Define by , then and . It is an embedding of into . So this is the sub-field isomorphic to .

(d) Take . Now . So has root in .

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