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In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field of rational numbers. Intuitively, it consists of ratios between integral domain elements.
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Ancient Greek mathematics, largely geometric in nature, drew on Egyptian numerical systems (especially Attic numerals), [9] with little interest in algebraic symbols, until the arrival of Diophantus of Alexandria, [10] who pioneered a form of syncopated algebra in his Arithmetica, which introduced symbolic manipulation of expressions. [11]
However, such an intermediate extension may exist if, for example, is a finite degree normal extension (in this case, K is the fixed field of the Galois group of E over F). Suppose that such an intermediate extension does exist, and [ E : F ] is finite, then [ S : F ] = [ E : K ] , where S is the separable closure of F in E . [ 18 ]
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations acting on their elements. [1] Algebraic structures include groups , rings , fields , modules , vector spaces , lattices , and algebras over a field .
Roland "Ron" Edwin Larson (born October 31, 1941) is a professor of mathematics at Penn State Erie, The Behrend College, Pennsylvania. [1] He is best known for being the author of a series of widely used mathematics textbooks ranging from middle school through the second year of college.
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