R[X] is never a field .... Sharp, Exercise 1.29 .... ....

In summary, Peter is looking for help with a proof for a theorem in Chapter 1 of the book Steps in Commutative Algebra by R. Y. Sharp. The theorem states that if a ring is not a field, then there exists a polynomial such that a_1 X (b_0 + b_1 X + \ ... \ ... \ + b_n X^n) = 1. However, the proof is impossible as the term on the right-hand side only has a term in X^0, while the left-hand side has terms in X in powers greater than 0. This confirms the theorem.
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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: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
 
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  • #2
Peter said:
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:I 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

The proof is correct.
 
  • #3
caffeinemachine said:
The proof is correct.
Thanks for confirming the proof caffeinemachine ... appreciate the help ...

Peter
 

Related to R[X] is never a field .... Sharp, Exercise 1.29 .... ....

What does "R[X] is never a field" mean?

"R[X] is never a field" is a statement in algebra, specifically in polynomial rings. It means that the polynomial ring R[X], which consists of all polynomials with coefficients in the ring R, can never be a field, which is a mathematical structure with certain properties such as having a multiplicative inverse for every non-zero element.

What is the significance of Sharp, Exercise 1.29 in relation to "R[X] is never a field"?

Sharp, Exercise 1.29 is a specific exercise in a mathematics textbook or course that deals with the concept of polynomial rings and fields. In this exercise, the author may ask the reader to prove or provide a counterexample for the statement "R[X] is never a field". This exercise is significant because it helps the reader better understand the concept and its implications.

Why is "R[X] is never a field" an important concept in algebra?

"R[X] is never a field" is an important concept in algebra because it helps to distinguish between different mathematical structures and their properties. It also highlights the limitations of polynomial rings and the importance of fields in certain mathematical contexts.

What are some examples of rings R for which "R[X] is never a field" holds true?

One example is the ring of integers Z. In this case, R[X] is the ring of polynomials with integer coefficients, and it is not a field because there is no multiplicative inverse for every non-zero polynomial. Another example is the ring of matrices over a field, where R[X] is the ring of polynomials with matrix coefficients.

How does the statement "R[X] is never a field" relate to other mathematical concepts?

The statement "R[X] is never a field" is closely related to other concepts in algebra, such as polynomial rings, fields, and ideals. It also has implications in other areas of mathematics, such as algebraic geometry and number theory.

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