Find th eminimal polynomial of sqrt2+sqrt3 over QQ

rocedwrp 2020-11-10 Answered
Find th eminimal polynomial of 2+3 over Q
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

Jayden-James Duffy
Answered 2020-11-11 Author has 91 answers
To find the smallest degree polynomial f(x) with coefficients in Q, such that f(2+3)=0.
The impo
rtant point is that the coefficients should all be in Q. For example, x(2+3) is not allowed, as its coefficients are not rational numbers.
Now, 3Q2.
So, we expect the minimal polynomial of 2+3 to have degree 4(,2+3Q(2)(3), a quadrantic extension of a quadrantic extension of Q.
We proceed to find this minimal polynomial by repeated squaring (to clear radicals and obtain rational coefficients)
x=2+3x2=3, squaring
(x2)2=3x222x+2=3, rearrange
x21=22x, square again
(x21)2=8x2x42x2+1=8x2, rearrange
Minimal polynomial is x410x2+1
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

New questions

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question