Polynomials in n indeterminates and UFDs

[Math Amateur]

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

 

[Opalg]

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.