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In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field of real numbers, a rational point is more commonly called a real point.
In mathematics, two non-zero real numbers a and b are said to be commensurable if their ratio a / b is a rational number; otherwise a and b are called incommensurable. (Recall that a rational number is one that is equivalent to the ratio of two integers.) There is a more general notion of commensurability in group theory.
This is a geometric take on Hilbert's irreducibility theorem which shows the rational numbers are Hilbertian. Results are applied to the inverse Galois problem . Thin sets (the French word is mince ) are in some sense analogous to the meagre sets (French maigre ) of the Baire category theorem .
There is a bijection between the set of k-rational points of |D| and the set of effective Weil divisors on X that are linearly equivalent to D. [1] The same definition is used if D is a Cartier divisor on a complete variety over k. [X/G] The quotient stack of, say, an algebraic space X by an action of a group scheme G. / /
The theorem extends to the other trigonometric functions as well. [2] For rational values of θ, the only rational values of the sine or cosine are 0, ±1/2, and ±1; the only rational values of the secant or cosecant are ±1 and ±2; and the only rational values of the tangent or cotangent are 0 and ±1. [3]
When working out a theorem, the use of this expression in the statement of the theorem indicates that the conditions involved may be not yet known to the speaker, and that the intent is to collect the conditions that will be found to be needed in order for the proof of the theorem to go through. the following are equivalent (TFAE)
Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties. [ 2 ] [ 3 ] In more abstract terms, arithmetic geometry can be defined as the study of schemes of finite type over the spectrum of the ring of integers .
Q – rational numbers. QED – "Quod erat demonstrandum" , a Latin phrase used at the end of a definitive proof. QEF – " Quod erat faciendum ", a Latin phrase sometimes used at the end of a geometrical construction.