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Primitive Pythagorean Triangle
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A triangle whose side lengths are a Pythagorean triple is a right triangle and called a Pythagorean triangle. A primitive Pythagorean triple is one in which , and are coprime (that is, they have no common divisor larger than 1). For example, is a primitive Pythagorean triple whereas is not. Every Pythagorean triple can be scaled to a unique primitive Pythagorean triple by dividing by their greatest common divisor. Conversely, every Pythagorean triple can be obtained by multiplying the elements of a primitive Pythagorean triple by a positive integer (the same for the three elements). The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula a^2+b^2=c^2; thus, Pythagorean triples describe the three integer s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Primitive Element (field Theory)
In field theory (mathematics), field theory, a simple extension is a field extension that is generated by the adjunction (field theory), adjunction of a single element, called a ''primitive element''. Simple extensions are well understood and can be completely classified. The primitive element theorem provides a characterization of the finite extension, finite simple extensions. Definition A field extension is called a simple extension if there exists an element in ''L'' with :L = K(\theta). This means that every element of can be expressed as a rational fraction in , with coefficients in ; that is, it is produced from and elements of by the field operations +, −, •, / . Equivalently, is the smallest field that contains both ' and . There are two different kinds of simple extensions (see below): # The element may be transcendental element, transcendental over , which means that it is not a zero of a function, root of any polynomial with coefficients in . In this case ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
