 #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
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