Pythagorean *-fields
نویسندگان
چکیده
منابع مشابه
Pythagorean Triples
Let n be a number. We say that n is square if and only if: (Def. 3) There exists m such that n = m2. Let us note that every number which is square is also natural. Let n be a natural number. Note that n2 is square. Let us observe that there exists a natural number which is even and square. Let us observe that there exists a natural number which is odd and square. Let us mention that there exist...
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The name comes from elementary geometry: if a right triangle has leg lengths x and y and hypotenuse length z, then x + y = z. Of course here x, y, z are positive real numbers. For most integer values of x and y, the integer x + y will not be a perfect square, so the positive real number √ x2 + y2 will be irrational: e.g. x = y = 1 =⇒ z = √ 2. However, a few integer solutions to x + y = z are fa...
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where B(v, w) = 12(Q(v + w) − Q(v) − Q(w)) is the bilinear form associated to Q. The transformation sw is linear, fixes the plane w⊥ = {v : v ⊥ w}, and acts by negation on the line through w. These properties characterize sw. We will use reflections associated to the four vectors e1 = (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1), and e1 + e2 + e3 = (1, 1, 1). The vectors e1, e2, and e3 form an ort...
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We call a field K semi-real closed if it is algebraically maximal with respect to a semi-ordering. It is proved that (as in the case of real closed fields) this is a Galois-theoretic property. We give a recursive description of all absolute Galois groups of semi-real closed fields of finite rank. I n t r o d u c t i o n By a well-known theorem of Artin and Schreier [AS], being a real closed fie...
متن کاملPythagorean powers of hypercubes
For n ∈ N consider the n-dimensional hypercube as equal to the vector space F2 , where F2 is the field of size two. Endow F2 with the Hamming metric, i.e., with the metric induced by the `1 norm when one identifies F2 with {0, 1} ⊆ R. Denote by `2 (F2 ) the n-fold Pythagorean product of F2 , i.e., the space of all x = (x1, . . . , xn) ∈ ∏n j=1 F n 2 , equipped with the metric ∀x, y ∈ n ∏ j=1 F2...
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ژورنال
عنوان ژورنال: Journal of Algebra
سال: 2000
ISSN: 0021-8693
DOI: 10.1006/jabr.1999.8149