Key to "Find the isomorphic ring with𝟞Z6x⟨x3−x⟩(Chinese remainder thm)"

Bradley Barron

Answered question

2022-04-14

Find the isomorphic ring with

Z_{6} \frac{x}{⟨ x^{3} - x ⟩}

(Chinese remainder thm)

libertydragonrbha

Beginner2022-04-15Added 15 answers

Step 1
Your first calculation fails because

x - 1

and

x + 1

are not co'.
Assume to the contrary that

x + 1

and

x - 1

are co' in

Z_{6} [x]

, then there are f,

g \in Z_{6} [X]

such that

1 = (x + 1) \cdot f + (x - 1) \cdot g

.
Plugging in

x = 1

, this yields

1 = 2 \cdot f (1)

,
which implies that 2 is invertible in

Z_{6}

, which is incorrect.

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