- #1

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## Homework Statement

I am reading R. Y. Sharp's book: "Steps in Commutative Algebra" Cambridge University Press (Second Edition) ... ...

I am focused on Chapter 1: Commutative Rings and Subrings ... ...

I need some help with Exercise 1.29 ...

Exercise 1.29 reads as follows:

## Homework Equations

Sharp's definitions, notation and remarks regarding R[X] are as follows:

## The Attempt at a Solution

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I am somewhat unsure about how to go about framing a valid and rigorous proof to demonstrate that ##R[X]## is never a field ...

But ... maybe the following is relevant ...

Consider ##a_1 X \in R[X]## ...

... then if ##R[X]## is a field ... there would be a polynomial ##b_0 + b_1 X + \ ... \ ... \ + b_n X^n## such that ...

... ##a_1 X ( b_0 + b_1 X + \ ... \ ... \ + b_n X^n ) = 1##

That is, we would require

##a_1 b_0 X + a_1 b_1 X^2 + \ ... \ ... \ + a_1 b_n X^{ n + 1} = 1## ... ... ... ... ... (1)

... But ... it is impossible for equation (1) to be satisfied as the term on the RHS has only a term in ##X^0## while the LHS only has terms in ##X## in powers greater than ##0## ...

Does the above qualify as a formal and rigorous proof ... if not ... what would constitute a formal and rigorous proof ...

Hope someone can help ...

Peter