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multiples of an algebraic number∗
pahio†
2013-03-21 19:18:19
Theorem. If α is an algebraic number, then there exists a non-zero multiple
of α which is an algebraic integer.
Proof. Let α be a root of the equation
xn +r1 xn−1 +r2 xn−2 + · · · +rn = 0,
where r1 , r2 , . . . , rn are rational numbers (n > 0). Let l be the least common
multiple of the denominators of the rj ’s. Then we have
0 = ln (αn+r1 αn−1+r2 αn−2+· · ·+rn ) = (lα)n+lr1 (lα)n−1+l2 r2 (lα)n−2+· · ·+ln rn ,
i.e. the multiple lα of α satisfies the algebraic equation
xn +lr1 xn−1 +l2 r2 xn−2 + · · · +ln rn = 0
with rational integer coefficients.
According to the theorem, any algebraic number ξ is a quotient of an algebraic integer (of the field Q(ξ)) and a rational integer.
∗ hMultiplesOfAnAlgebraicNumberi
created: h2013-03-21i by: hpahioi version: h37126i
Privacy setting: h1i hTheoremi h11R04i
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