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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.
In terms of its expression as a determinant of a (2n − 1) × (2n − 1) matrix (the Sylvester matrix) divided by a n, the determinant is homogeneous of degree 2n − 1 in the entries, and dividing by a n makes the degree 2n − 2. The discriminant of a polynomial of degree n is homogeneous of degree n(n − 1) in the roots.
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 ...
Since the number of integral ideals of given norm is finite, the finiteness of the class number is an immediate consequence, [1] and further, the ideal class group is generated by the prime ideals of norm at most M K. Minkowski's bound may be used to derive a lower bound for the discriminant of a field K given n, r 1 and r 2.
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 following table shows some orders of small discriminant of quadratic fields. The maximal order of an algebraic number field is its ring of integers, and the discriminant of the maximal order is the discriminant of the field. The discriminant of a non-maximal order is the product of the discriminant of the corresponding maximal order by the ...
By the 5th grade, at the red-brick Hamilton Avenue School in nearby Greenwich, he’d published three poems in the school newspaper. One, written after a class lecture about drinking and driving, described the thoughts of a driver as he was dying in a car crash.
In a covering map the Euler–Poincaré characteristic should multiply by the number of sheets; ramification can therefore be detected by some dropping from that. The z → z n mapping shows this as a local pattern: if we exclude 0, looking at 0 < |z| < 1 say, we have (from the homotopy point of view) the circle mapped to itself by the n-th power map (Euler–Poincaré characteristic 0), but ...
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