Polynomials in n indeterminates and UFDs

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SUMMARY

The polynomial ring $$K[ X_1, X_2, \ldots, X_n]$$ over a field $$K$$ is established as a unique factorization domain (UFD). This conclusion is derived from the theorem stating that if a ring $$R$$ is a UFD, then the polynomial ring $$R[X]$$ is also a UFD. The proof utilizes induction on the number of indeterminates, demonstrating that $$K[ X_1, X_2, \ldots, X_n]$$ can be expressed as $$ (K[ X_1, X_2, \ldots, X_{n-1}])[X_n]$$. A detailed discussion and proof of this theorem can be found in the referenced materials.

PREREQUISITES
  • Understanding of unique factorization domains (UFDs)
  • Familiarity with polynomial rings
  • Knowledge of induction proofs in algebra
  • Basic concepts of field theory
NEXT STEPS
  • Study the properties of unique factorization domains in detail
  • Learn about polynomial rings and their structure
  • Explore induction proofs in algebraic contexts
  • Read "Introduction to Plane Algebraic Curves" by Ernst Kunz for deeper insights
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Mathematicians, algebraists, and students studying abstract algebra, particularly those interested in unique factorization domains and polynomial rings.

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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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