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Field (Mathematics)
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The concept of a field is due to Dedekind, who used the word Körper "body" for this notion. He ... was the first to define rings (then called order or order-modul), but the term "a ring" (Zahlring) was invented by Hilbert. [1]
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In the formalist view, widely accepted as a description by professionals in the field, mathematics is defined as the investigation of axiomatically defined abstract structures using symbolic logic and mathematical notation. Mathematics might accordingly be seen as an extension of spoken and written natural languages, with an extremely precisely defined vocabulary and grammar, for the purpose of describing and exploring physical and conceptual relationships. There are other views, and some are described in the article on the philosophy of mathematics.
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For a given field F, the set F(X) of rational functions in the variable X with coefficients in F is a field; this is the quotient field of the ring of polynomials F[X]. This is the simplest example of a transcendental extension of F.
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The nimbers form a field, again except for the fact that they are a proper class. The set of nimbers with birthday smaller than 2^(2^n), the nimbers with birthday smaller than any infinite cardinal are all examples of fields.
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Mathematics is the King of Science, and there is no lack of tools to assist you in solving the mathematical problems you face in your particular field. Whether you're an actuary working on pricing insurance policies or a celestial navigator determining the optimal path for flying a spacecraft to Pluto, programs are available to help with your quests for answers.
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There is (up to isomorphism) exactly one finite field with q elements, for every finite number q which is a power of a prime number, q≠ 1. (No finite field can exist with any other number of elements.) This is usually denoted Fq . Finite fields are ... called Galois fields.
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Field (Mathematics)