How can I show these polynomials are not co'?x, x−1 and x+1 in the ring Z6[x]

Step 1The ideal (x−1, x+1) equals the ideal(2, x−1) and this is not the whole ring, otherwise 1∈(2, x−1), so 1=2f(x)+(x−1)g(x).Now send x to 1 and get a contradiction.

"How can I show these polynomials are not..." solved and explained

College
Algebra
Abstract algebra

Paula Good

Answered question

2022-03-25

How can I show these polynomials are not co'?

x, x - 1

and

x + 1

in the ring

Z_{6} [x]

Answer & Explanation

Aarlenlsi1

Beginner2022-03-26Added 10 answers

Step 1
The ideal

(x - 1, x + 1)

equals the ideal

(2, x - 1)

and this is not the whole ring, otherwise

1 \in (2, x - 1)

, so

1 = 2 f (x) + (x - 1) g (x)

.
Now send x to 1 and get a contradiction.

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