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... The additional points of Spec(R) are valuable in many situations and a systematic study of them leads to the general notion of schemes. As just one example, the classical Bezout’s theorem is only valid for algebraically closed fields, but admits a scheme–theoretic generalization which holds over non ...
... The additional points of Spec(R) are valuable in many situations and a systematic study of them leads to the general notion of schemes. As just one example, the classical Bezout’s theorem is only valid for algebraically closed fields, but admits a scheme–theoretic generalization which holds over non ...
Introducing Algebraic Number Theory
... In this chapter, unless otherwise specified, all rings are assumed commutative. Let A be a subring of the ring R, and let x ∈ R. We say that x is integral over A if x is a root of a monic polynomial f with coefficients in A. The equation f (X) = 0 is called an equation of integral dependence for x over ...
... In this chapter, unless otherwise specified, all rings are assumed commutative. Let A be a subring of the ring R, and let x ∈ R. We say that x is integral over A if x is a root of a monic polynomial f with coefficients in A. The equation f (X) = 0 is called an equation of integral dependence for x over ...
Transcendental extensions
... k(X1 , · · · , Xn ) = Qk[X1 , · · · , Xn ] Here the capital letters Xi are formal variables. So, k[X1 , · · · , Xn ] is the ring of polynomials in the generators X1 , · · · , Xn with coefficients in the field k and Q is the functor which inverts all the nonzero elements. I.e., QR is the quotient fie ...
... k(X1 , · · · , Xn ) = Qk[X1 , · · · , Xn ] Here the capital letters Xi are formal variables. So, k[X1 , · · · , Xn ] is the ring of polynomials in the generators X1 , · · · , Xn with coefficients in the field k and Q is the functor which inverts all the nonzero elements. I.e., QR is the quotient fie ...
Purely Algebraic Results in Spectral Theory
... Let (r , S) be a maximal resolvent family in A and let J be a two-sided ideal in A and let πJ : A → A/J be the natural quotient map. Then one can define the resolvent family (rJ , S) in A/J by rJ ,λ = πJ (rλ ). However (rJ , S) may not be maximal, so one should construct its maximal extension (r̃J , ...
... Let (r , S) be a maximal resolvent family in A and let J be a two-sided ideal in A and let πJ : A → A/J be the natural quotient map. Then one can define the resolvent family (rJ , S) in A/J by rJ ,λ = πJ (rλ ). However (rJ , S) may not be maximal, so one should construct its maximal extension (r̃J , ...