单词	constructible number
释义	constructible number (constructible point) A point in the plane is constructible from two given points, if it can be drawn using a construction with ruler and compass. Assigning the two points coordinates (0,0) and (1,0), a real number α is constructible if the point (α,0) is constructible. The constructible numbers form a field, which contains the rational numbers and is contained in the algebraic numbers. However, not every algebraic number is constructible. Because a construction involves, at each step, intersecting lines and circles, algebraically nothing more complicated than a quadratic equation is being solved. Consequently (by the tower law), it follows that if α is constructible, then the degree of the field extension ℚ(α): ℚ is a power of 2. (The converse is false.) Thus, , though algebraic, is not constructible as has degree 3; moreover, this fact means it is impossible to duplicate the cube. As trisecting the angle, 60° would be equivalent, algebraically to solving the cubic t3–3t–1=0. As this cubic is irreducible, ℚ(t):ℚ has degree 3 for any root, and so it is impossible to trisect the angle 60°. Pierre Wantzel demonstrated these facts in 1837 and also showed that if a regular n-gon is constructible, then n is a product of powers of 2 and distinct Fermat primes, the converse of a famous result of Gauss. The impossibility of squaring the circle would become clear in 1882, when it was shown that pi is transcendental by Lindemann. See construction with ruler and compass.
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