Unique factorization over fields/rings

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SUMMARY

The discussion centers on the definition of divisibility in the context of polynomials within commutative rings, specifically addressing a potential typo in the notation used in the textbook. The quote states that if R is a commutative ring, a polynomial d in R[x] is a divisor of f in R[x] if f = qd for some q in F[x]. Participants question whether q should be in R[x] instead of F[x], as F[x] was not previously defined. The consensus suggests that the notation may indeed be incorrect, indicating a need for clarification in the textbook.

PREREQUISITES
  • Understanding of commutative rings
  • Familiarity with polynomial rings, specifically R[x]
  • Knowledge of field theory and polynomial fields, such as F[x]
  • Basic concepts of divisibility in algebra
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  • Research the properties of commutative rings and their polynomial rings
  • Study the definitions and examples of divisibility in polynomial rings
  • Explore the differences between R[x] and F[x] in algebraic contexts
  • Examine common notational conventions in algebra textbooks
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Students of abstract algebra, mathematicians focusing on ring theory, and educators seeking clarity on polynomial divisibility concepts.

PsychonautQQ
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Here is a direct quote from my textbook:
If R is a commutative ring, we say that a polynomial d in R[x] is a divisor of f in R[x] if f = qd for some q in F[x].

My question is did they mean to put q in F[x}? q isn't in R[x]? They didn't mention F[x] before this, is F[x] the field of all polynomials or something?
 
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PsychonautQQ said:
is F[x] the field of all polynomials or something?

To me, the "F[x]" looks like a typo that should be "R[x]" instead.
 
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i agree.
 
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