How can I show these polynomials are not coprime? x,\ x-1

Malachi Novak 2022-04-24 Answered

How can I show these polynomials are not co'?

x, x - 1

and

x + 1

in the ring

Z_{6} [x]

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Answers (1)

veceritzpzg

Answered 2022-04-25 Author has 16 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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