Unique factorization over fields/rings

In summary, the conversation discusses the definition of a divisor in a commutative ring and the potential typo of using "F[x]" instead of "R[x]."
  • #1
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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  • #2
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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  • #3
i agree.
 
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1. What is unique factorization over fields/rings?

Unique factorization over fields/rings is a mathematical concept that refers to the ability to express any non-zero element as a product of irreducible elements in only one way. This means that there are no other ways to break down a number into smaller factors besides the one unique way.

2. Why is unique factorization important?

Unique factorization is important because it allows for the simplification of complex numbers and equations. It also plays a crucial role in many areas of mathematics, including number theory and abstract algebra.

3. What is the difference between unique factorization over fields and rings?

The main difference between unique factorization over fields and rings is the type of elements that are involved. In fields, all non-zero elements have a unique factorization, while in rings, only some elements have a unique factorization. Additionally, in fields, every element can be inverted, while in rings, only some elements have multiplicative inverses.

4. Can unique factorization fail in certain cases?

Yes, unique factorization can fail in certain cases, such as in polynomial rings over non-algebraically closed fields. In these cases, it is possible for an element to have multiple factorizations into irreducible elements. This is known as non-unique factorization.

5. How is unique factorization related to prime factorization?

Unique factorization and prime factorization are closely related concepts. Prime factorization refers to the decomposition of a number into its prime factors, while unique factorization refers to the decomposition of a number into its irreducible factors. In unique factorization, prime factors are considered to be a special case of irreducible factors.

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