Assume that " X " is a normal integral separated scheme of finite type over a field.
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1 ) The Concept-With Normal integrals, we are trying to find an area under a curve.
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A classical result, based on a result of David Hilbert, is that a tamely ramified abelian number field has a normal integral basis.
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In cases where the theorem states that a normal integral basis does exist, such a basis may be constructed by means of Gaussian periods.
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For example if we take a prime number, has a normal integral basis consisting of all the-th roots of unity other than.
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:Each finite tamely ramified abelian extension of a fixed number field has a relative normal integral basis if and only if "'Q "'} }.
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From the point of view of algebraic number theory it is of interest to study " normal integral bases ", where we try to replace " L " and " K " by the rings of algebraic integers they contain.
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A formal power series version of Zariski's main theorem says that if " x " is a normal point of a variety then it is analytically normal; in other words the completion of the local ring at " x " is a normal integral domain.
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In fact all the subfields of the cyclotomic fields for " p "-th roots of unity for " p " a " prime number " have normal integral bases ( over "'Z "'), as can be deduced from the theory of Gaussian periods ( the Hilbert Speiser theorem ).
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Everest came from a working-class family and studied at Bedford College ( now Royal Holloway College ) of the University of London : he took a Ph . D . in 1983 under the supervision of Colin J . Bushnell of King's College London ( " The distribution of normal integral generators in tame extensions of Q . " ) He joined the faculty of the University of East Anglia in 1983 as a lecturer and spent his academic career there.
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