Find th eminimal polynomial of sqrt2+sqrt3 over QQ



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Find th eminimal polynomial of 2+3 over Q

Answer & Explanation

Jayden-James Duffy

Jayden-James Duffy

Skilled2020-11-11Added 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

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