In algebra, a Pythagorean field is a field in which every sum of two squares is a square: equivalently it has Pythagoras number equal to 1. A Pythagorean extension of a field is an extension obtained by adjoining an element for some in . So a Pythagorean field is one closed under taking Pythagorean extensions. For any field there is a minimal Pythagorean field containing it, unique up to isomorphism, called its Pythagorean closure.[1] The Hilbert field is the minimal ordered Pythagorean field.[2]
algebra, a Pythagoreanfield is a field in which every sum of two squares is a square: equivalently it has Pythagoras number equal to 1. A Pythagorean extension...
Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans...
A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), a well-known...
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle...
known as a Pythagorean box. Pythagorean expectation – a method of statistical analysis inspired by the Pythagorean theorem Pythagoreanfield – in algebra...
such that every sum of squares in K is a sum of p squares. A Pythagoreanfield is a field with Pythagoras number 1: that is, every sum of squares is already...
conversely (for example, R is Pythagorean); however, every non-formally real Pythagoreanfield is quadratically closed. A field is quadratically closed if...
ancient Ionian Greek philosopher, polymath and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graecia...
Euclidean closure of the rational numbers. Every Euclidean field is an ordered Pythagoreanfield, but the converse is not true. If E/F is a finite extension...
Hilbert field may refer to: The Hilbert field, the minimal ordered Pythagoreanfield A Hilbert field is one with minimal Kaplansky radical Hilbert class...
A Pythagorean prime is a prime number of the form 4 n + 1 {\displaystyle 4n+1} . Pythagorean primes are exactly the odd prime numbers that are the sum...
exponential field is a Euclidean field. Consequently, every ordered exponential field is an ordered Pythagoreanfield. Not every real-closed field is a formally...
non-Archimedean ordered Pythagoreanfield Ω(t), a Pythagorean closure of the field of rational functions R(t), consisting of the smallest field of real-valued...
whose signatures are zero; W has Krull dimension 1. If k is a real Pythagoreanfield then the zero-divisors of W are the elements for which some signature...
theorem. Pythagoras established the Pythagorean School, which is credited with the first proof of the Pythagorean theorem, though the statement of the...
of music. The theory, originating in ancient Greece, was a tenet of Pythagoreanism, and was later developed by 16th-century astronomer Johannes Kepler...
Chinese mathematics, but, unlike that of the Pythagoreans, it seems to have led nowhere. Like the Pythagoreans' perfect numbers, magic squares have passed...
tagged that, too. — Bill Simmons, sportswriter, Grantland Pythagorean expectation, or Pythagorean projection, calculates a percentage based on the number...
both the Egyptians and the Babylonians were aware of versions of the Pythagorean theorem about 1500 years before Pythagoras and the Indian Sulba Sutras...
important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level...
before the Pythagoreans... It is evident, however, that at this early date the Babylonians not only had completely mastered the Pythagorean problem, but...
parallelograms (35–45); and the Pythagorean theorem (46–48). The last of these includes the earliest surviving proof of the Pythagorean theorem, described by Sialaros...
manner are less than two right angles." (Book I proposition 17) and the Pythagorean theorem "In right-angled triangles the square on the side subtending...
the smallest integer-sided right triangle, making part of the smallest Pythagorean triple (3, 4, 5). In geometry, the regular five-sided pentagon is the...
value of x The other extension of IFS is what is known as Pythagorean fuzzy sets. Pythagorean fuzzy sets are more flexible than IFSs. IFSs are based on...
circa 1650 BCE. Historians of Mesopotamia have confirmed that use of the Pythagorean rule dates back to the Old Babylonian Empire (20th–16th centuries BC)...
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