Elliptic curves are now normally studied in some variant of Weierstrass's elliptic functions, defining a quadratic extension of the field of rational functions made by extracting the square root of a cubic.
12.
These are all products of the Vandermonde polynomial and a symmetric polynomial, and form a quadratic extension of the ring of symmetric polynomials : the Vandermonde polynomial is a square root of the discriminant.
13.
It is not possible to tell whether \ mathbb Z / N \ mathbb Z [ \ sqrt t ] is actually a quadratic extension of " N " without knowing the factorisation.
14.
The characterization is the following : a complex number is constructible if and only if it lies in a field at the top of a finite tower of quadratic extensions, starting with the rational field.
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*PM : prime ideal decomposition in quadratic extensions of \ mathbb { Q }, id = 4643-- WP guess : prime ideal decomposition in quadratic extensions of \ mathbbQ-- Status:
16.
*PM : prime ideal decomposition in quadratic extensions of \ mathbb { Q }, id = 4643-- WP guess : prime ideal decomposition in quadratic extensions of \ mathbbQ-- Status:
17.
The argument given does not apply in this case, because some of the endomorphisms of supersingular elliptic curves are only defined over a quadratic extension of the field of order " p " .)
18.
Since the field of constructible points is closed under " square roots ", it contains all points that can be obtained by a finite sequence of quadratic extensions of the field of complex numbers with rational coefficients.
19.
*PM : proof of prime ideal decomposition in quadratic extensions of \ mathbb { Q }, id = 8002 new !-- WP guess : proof of prime ideal decomposition in quadratic extensions of \ mathbbQ-- Status:
20.
*PM : proof of prime ideal decomposition in quadratic extensions of \ mathbb { Q }, id = 8002 new !-- WP guess : proof of prime ideal decomposition in quadratic extensions of \ mathbbQ-- Status:
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