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Moreover, if one sets x = 1 + t, one gets without computation that () = (+) is a polynomial in t with the same first coefficient 3 and constant term 1. [2] The rational root theorem implies thus that a rational root of Q must belong to {,}, and thus that the rational roots of P satisfy = + {,,,}.
An element a of F is integral over R if it is a root of a monic polynomial with coefficients in R. A complex number that is integral over the integers is called an algebraic integer. This terminology is motivated by the fact that the integers are exactly the rational numbers that are also algebraic integers.
Rational root theorem – Relationship between the rational roots of a polynomial and its extreme coefficients; Geometrical properties of polynomial roots – Geometry of the location of polynomial roots; Gauss–Lucas theorem – Geometric relation between the roots of a polynomial and those of its derivative
While the proofs by infinite descent are constructively valid when "irrational" is defined to mean "not rational", we can obtain a constructively stronger statement by using a positive definition of "irrational" as "quantifiably apart from every rational". Let a and b be positive integers such that 1< a / b < 3/2 (as 1<2< 9/4 satisfies ...
In the case of two nested square roots, the following theorem completely solves the problem of denesting. [2]If a and c are rational numbers and c is not the square of a rational number, there are two rational numbers x and y such that + = if and only if is the square of a rational number d.
By the rational root theorem, this has no rational zeroes. Neither does it have linear factors modulo 2 or 3. The Galois group of f(x) modulo 2 is cyclic of order 6, because f(x) modulo 2 factors into polynomials of orders 2 and 3, (x 2 + x + 1)(x 3 + x 2 + 1). f(x) modulo 3 has no linear or quadratic factor, and hence is irreducible. Thus its ...
Rearranging gives 8x 3 − 6x − 1 = 0, which fails the rational root test as none of the rational numbers suggested by the theorem is actually a root. Therefore, the minimal polynomial of cos(20°) has degree 3, whereas the degree of the minimal polynomial of any constructible number must be a power of two.
The numbers and are algebraic since they are roots of polynomials x 2 − 2 and 8x 3 − 3, respectively. The golden ratio φ is algebraic since it is a root of the polynomial x 2 − x − 1. The numbers π and e are not algebraic numbers (see the Lindemann–Weierstrass theorem). [3]