Polynomials in n indeterminates and UFDs

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In summary, Ernst Kunz states in the introduction to Chapter 1 of his book "Introduction to Plane Algebraic Curves" that the polynomial ring K[ X_1, X_2, \ ... \ ... \ X_n] over a field K is a unique factorization domain, but he does not provide a proof. However, there is a theorem that states if R is a UFD, then so is R[X], and this result can be proven by induction on the number of indeterminates. A discussion and proof of this theorem can be found in the provided link.
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In the introduction to Chapter 1 of his book "Introduction to Plane Algebraic Curves", Ernst Kunz states that the polynomial ring \(\displaystyle K[ X_1, X_2, \ ... \ ... \ X_n]\) over a field \(\displaystyle K\) is a unique factorization domain ... ... but he does not prove this fact ...

Can someone demonstrate a proof of this proposition ... or point me to a text or online notes that contain a proof ...

Help will be appreciated ... ...

Peter
 
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Peter said:
In the introduction to Chapter 1 of his book "Introduction to Plane Algebraic Curves", Ernst Kunz states that the polynomial ring \(\displaystyle K[ X_1, X_2, \ ... \ ... \ X_n]\) over a field \(\displaystyle K\) is a unique factorization domain ... ... but he does not prove this fact ...

Can someone demonstrate a proof of this proposition ... or point me to a text or online notes that contain a proof ...

Help will be appreciated ... ...

Peter
There is a theorem that if $R$ is a UFD then so is $R[X]$. Your result then follows by induction on the number of indeterminates, because $K[ X_1, X_2, \ldots, X_n] = (K[ X_1, X_2, \ldots, X_{n-1}])[X_n]$.

There is a discussion and proof of that theorem here.
 

FAQ: Polynomials in n indeterminates and UFDs

What are polynomials in n indeterminates?

Polynomials in n indeterminates are expressions that consist of one or more variables raised to various powers and multiplied by coefficients. The number of variables, or indeterminates, can vary and is typically denoted by the letter n.

What is a UFD?

UFD stands for Unique Factorization Domain. It is a type of mathematical structure in which every element can be expressed as a unique product of irreducible elements, similar to how every integer can be expressed as a unique product of prime numbers.

How are polynomials in n indeterminates and UFDs related?

Polynomials in n indeterminates are often studied within the context of UFDs, as UFDs provide a useful framework for understanding and manipulating polynomials. In particular, the unique factorization property of UFDs allows for the simplification and manipulation of polynomials.

What are some examples of polynomials in n indeterminates?

Some examples of polynomials in n indeterminates include: x^2 + 2x + 1 (in one indeterminate, x), xy^2 + 3y + 5 (in two indeterminates, x and y), and x^3y^2z^4 + 7x^2y^3z + 4 (in three indeterminates, x, y, and z).

What is the significance of studying polynomials in n indeterminates and UFDs?

Studying polynomials in n indeterminates and UFDs is important in various mathematical fields, such as algebra, number theory, and geometry. It also has practical applications in areas such as computer science and cryptography. Understanding these concepts can lead to a deeper understanding of mathematics as a whole.

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