MHB GCD is same in a field and its superfield.

  • Thread starter Thread starter caffeinemachine
  • Start date Start date
  • Tags Tags
    Field Gcd
AI Thread Summary
In an extension field K of a field F, the monic greatest common divisor (GCD) of polynomials p(t) and q(t) in F[t] is identical to their monic GCD in K[t]. If p(t) and q(t) share a non-trivial common factor in K[t], it implies they must also have one in F[t], contradicting the assumption that they do not. This leads to the conclusion that any common factor in K must also exist in F. Therefore, the GCD remains unchanged when moving from F[t] to K[t]. The relationship between the GCDs in both fields highlights the consistency of polynomial factorization across field extensions.
caffeinemachine
Gold Member
MHB
Messages
799
Reaction score
15
Let $K$ be an extension field of a field $F$ and let $p(t),q(t)\in F[t]$. Show that the monic greatest common divisors of $p(t)$ and $q(t)$ in $F[t]$ is same as the monic greatest common divisor of $p(t)$ and $q(t)$ in $K$.
 
Mathematics news on Phys.org
Hint:

Let $p(t)$ and $q(t)$ have a non trivial common factor in $K[t]$. Assume that $p$ and $q$ don't have a non-trivial common factor in
$F[t]$. Then there exist $a,b\in F[t]$ such that $pa+qb=1$. But this contradicts the fact that $p$ and $q$ have a non-trivial common factor in $K[t]$.
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top