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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It is an integer, and is an invariant property of the field " F ", not depending on the choice of integral basis.
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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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If is a square-free integer and ) } } is the corresponding quadratic field, then is a ring of quadratic integers and its integral basis is given by ) / 2 ) } } if and by ) } } if.
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In the other direction, given a pair ( X, \ omega ) where X is a compact Riemann surface and \ omega a holomorphic 1-form one can construct a polygon by using the complex numbers \ int _ { \ gamma _ j } \ omega where \ gamma _ j are disjoint paths between the zeroes of \ omega which form an integral basis for the relative cohomology.
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