MHB Polynomials in n indeterminates and UFDs

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The polynomial ring K[X1, X2, ..., Xn] over a field K is established as a unique factorization domain (UFD) based on a theorem stating that if R is a UFD, then R[X] is also a UFD. This property can be demonstrated through induction on the number of indeterminates, showing that K[X1, X2, ..., Xn] can be expressed as (K[X1, X2, ..., Xn-1])[Xn]. A reference to a discussion and proof of this theorem is provided for further clarification. The conversation emphasizes the need for a formal proof or additional resources to support this assertion.
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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 $$K[ X_1, X_2, \ ... \ ... \ X_n]$$ over a field $$K$$ 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 $$K[ X_1, X_2, \ ... \ ... \ X_n]$$ over a field $$K$$ 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.
 
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