Field axioms
References
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Field Axioms Media
The regular heptagon cannot be constructed using only a straightedge and compass construction; this can be proven using the field of constructible numbers.
The multiplication of complex numbers can be visualized geometrically by rotations and scalings.
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The geometric mean theorem asserts that h2 = pq. Choosing q = 1 allows construction of the square root of a given constructible number p.
In modular arithmetic modulo 12, 9 + 4 = 1 since 9 + 4 = 13 in Z, which divided by 12 leaves remainder 1. However, Z/12Z is not a field because 12 is not a prime number.
Each bounded real set has a least upper bound.
The sum of three points P, Q, and R on an elliptic curve E (red) is zero if there is a line (blue) passing through these points.
A compact Riemann surface of genus two (two handles). The genus can be read off the field of meromorphic functions on the surface.
The fifth roots of unity form a regular pentagon.
Weisstein, Eric W.. Field Axioms (in en). mathworld.wolfram.com. Retrieved 2021-09-15.
- ↑ Apostol, T. M. "The Field Axioms." §I 3.2 in Calculus, 2nd ed., Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra. Waltham, MA: Blaisdell, pp. 17-19, 1967.
