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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.
For a concrete example one can take R = Z[i√5], p = 1 + i√5, a = 1 − i√5, q = 2, b = 3. In this example the polynomial 3 + 2 X + 2 X 2 (obtained by dividing the right hand side by q = 2 ) provides an example of the failure of the irreducibility statement (it is irreducible over R , but reducible over its field of fractions Q [ i √5] ).
The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, ... The rational root theorem ...
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
Vieta's formulas relate the polynomial coefficients to signed sums of products of the roots r 1, r 2, ... Properties of polynomial roots; Rational root theorem;
Van der Waerden [11] cites the polynomial f(x) = x 5 − x − 1. 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).
The square root of the Gelfond–Schneider constant is the transcendental number = 1.632 526 919 438 152 844 77.... This same constant can be used to prove that "an irrational elevated to an irrational power may be rational", even without first proving its transcendence.