SUMMARY
The monic greatest common divisor (GCD) of two polynomials $p(t)$ and $q(t)$ in a field $F[t]$ is identical to the monic GCD of the same polynomials in an extension field $K$. This conclusion is reached by assuming the existence of a non-trivial common factor in $K[t]$ while $p(t)$ and $q(t)$ lack such a factor in $F[t]$. The contradiction arising from the existence of coefficients $a, b \in F[t]$ satisfying $pa + qb = 1$ confirms that the GCD remains unchanged across the fields.
PREREQUISITES
- Understanding of polynomial rings, specifically $F[t]$ and $K[t]$.
- Knowledge of the concept of greatest common divisors in the context of polynomials.
- Familiarity with field extensions and their properties.
- Basic algebraic manipulation skills involving polynomials.
NEXT STEPS
- Study the properties of polynomial rings over fields, focusing on $F[t]$ and $K[t]$.
- Learn about the Euclidean algorithm for finding GCDs of polynomials.
- Explore the implications of field extensions on algebraic structures.
- Investigate examples of non-trivial common factors in polynomial rings.
USEFUL FOR
Mathematicians, algebraists, and students studying abstract algebra, particularly those interested in polynomial theory and field extensions.