Yesterday, 09:04 PM
Gauss's lemma
Summary
Gauss's lemma for polynomials is a fundamental algebraic theorem stating that the product of two primitive polynomials—those whose coefficients share a greatest common divisor of 1—is also primitive. Originating in Carl Friedrich Gauss's 1801 treatise Disquisitiones Arithmeticae, the lemma holds for polynomials over the integers and extends to any unique factorization domain (UFD). Its primary corollary proves that a non-constant primitive polynomial is irreducible over an integral domain (like the integers) if and only if it is irreducible over its field of fractions (like the rational numbers). This key result guarantees that polynomial rings over UFDs are themselves UFDs, forming the core theoretical foundation for modern polynomial factorization and greatest common divisor algorithms.
ARTICE
Summary
Gauss's lemma for polynomials is a fundamental algebraic theorem stating that the product of two primitive polynomials—those whose coefficients share a greatest common divisor of 1—is also primitive. Originating in Carl Friedrich Gauss's 1801 treatise Disquisitiones Arithmeticae, the lemma holds for polynomials over the integers and extends to any unique factorization domain (UFD). Its primary corollary proves that a non-constant primitive polynomial is irreducible over an integral domain (like the integers) if and only if it is irreducible over its field of fractions (like the rational numbers). This key result guarantees that polynomial rings over UFDs are themselves UFDs, forming the core theoretical foundation for modern polynomial factorization and greatest common divisor algorithms.
ARTICE
┌────────────────────────────────┐
│ KONSTANTINOS MICHAILIDIS │
└────────────────────────────────┘
│ KONSTANTINOS MICHAILIDIS │
└────────────────────────────────┘