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

I need some help with Exercise 1.29 ...

Exercise 1.29 reads as follows:View attachment 8169I am somewhat unsure about how to go about framing a valid and rigorous proof to demonstrate that \(\displaystyle R[X]\) is never a field ...

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

Consider \(\displaystyle a_1 X \in R[X]\) ...

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

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

That is, we would require

\(\displaystyle a_1 b_0 X + a_1 b_1 X^2 + \ ... \ ... \ + a_1 b_n X^{ n + 1} = 1\) ... ... ... ... ... (1) ... But ... it is impossible for equation (1) to be satisfied as the term on the RHS has only a term in \(\displaystyle X^0\) while the LHS only has terms in \(\displaystyle X\) in powers greater than \(\displaystyle 0\) ...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