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Then the discriminant is the product of the a i, which is well-defined as a class in K/(K ×) 2. Geometrically, the discriminant of a quadratic form in three variables is the equation of a quadratic projective curve. The discriminant is zero if and only if the curve is decomposed in lines (possibly over an algebraically closed extension of the ...
The discriminant of K is 49 = 7 2. Accordingly, the volume of the fundamental domain is 7 and K is only ramified at 7. In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the (ring of integers of the) algebraic number field.
For this converse the field discriminant is needed. This is the Dedekind discriminant theorem. In the example above, the discriminant of the number field () with x 3 − x − 1 = 0 is −23, and as we have seen the 23-adic place ramifies. The Dedekind discriminant tells us it is the only ultrametric place that does.
The symbolic representation of the discriminant is = where a and b are the symbols. The meaning of the expression (ab) 2 is as follows. First of all, (ab) is a shorthand form for the determinant of a matrix whose rows are a 1, a 2 and b 1, b 2, so
By definition, the different ideal δ K is the inverse fractional ideal I −1: it is an ideal of O K. The ideal norm of δ K is equal to the ideal of Z generated by the field discriminant D K of K. The different of an element α of K with minimal polynomial f is defined to be δ(α) = f′(α) if α generates the field K (and zero otherwise ...
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Reassuring global firms that Bangladesh remains open for business is a top priority of the interim government. Still, the glacial pace of reform means doubts fester. Muhammad Yunus on the Race to ...
Similarly, 2 + √-5 and 2 - √-5 divide the product 3 2, but neither of these elements divides 3 itself, so neither of them are prime. As there is no sense in which the elements 3, 2 + √-5 and 2 - √-5 can be made equivalent, unique factorization fails in Z[√-5]. Unlike the situation with units, where uniqueness could be repaired by ...