
Hilbert`s Nullstellensatz and the Beginning of Algebraic Geometry
... n = 1 is easy, because for single variable polynomial rings k[X], one can divide one polynoinial f by another polynomial 9 of degree deg 9 = d and get a remainder T such that either T = 0 or deg T < d. The proof given in (ii) of example 2.3 above to show that all ideals in Z are principal (singly ge ...
... n = 1 is easy, because for single variable polynomial rings k[X], one can divide one polynoinial f by another polynomial 9 of degree deg 9 = d and get a remainder T such that either T = 0 or deg T < d. The proof given in (ii) of example 2.3 above to show that all ideals in Z are principal (singly ge ...
Universal Identities I
... for indeterminates A, B, C, A0 , B 0 , and C 0 and f, g, and h in Z[A, B, C, A0 , B 0 , C 0 ]. Notice (2.1) implies a similar formula for sums of three squares in any commutative ring by specializing the 6 indeterminates to any 6 elements of any commutative ring. So (2.1) implies that sums of three ...
... for indeterminates A, B, C, A0 , B 0 , and C 0 and f, g, and h in Z[A, B, C, A0 , B 0 , C 0 ]. Notice (2.1) implies a similar formula for sums of three squares in any commutative ring by specializing the 6 indeterminates to any 6 elements of any commutative ring. So (2.1) implies that sums of three ...
Applications of Logic to Field Theory
... algebraically closed fields of characteristic zero and cardinality κ are isomorphic. Similarly, for each prime p, ACF ∪ {ψp } is uncountably categorical. Thus, for example, C is the unique (up to isomorphism) algebraically closed field of characteristic zero having cardinality 2ℵ0 . To see that ACF ...
... algebraically closed fields of characteristic zero and cardinality κ are isomorphic. Similarly, for each prime p, ACF ∪ {ψp } is uncountably categorical. Thus, for example, C is the unique (up to isomorphism) algebraically closed field of characteristic zero having cardinality 2ℵ0 . To see that ACF ...
Inversion Modulo Zero-dimensional Regular Chains
... invertible, and if so, compute its inverse. We are also interested the generalization to matrices over RT : given a (d × d) matrix A ∈ Md (RT ), decide whether it is invertible, and if so, compute its inverse. We simply call this the problem of invertibility test / inversion in RT (or in Md (RT )). ...
... invertible, and if so, compute its inverse. We are also interested the generalization to matrices over RT : given a (d × d) matrix A ∈ Md (RT ), decide whether it is invertible, and if so, compute its inverse. We simply call this the problem of invertibility test / inversion in RT (or in Md (RT )). ...
Algebra Expressions and Real Numbers
... 3) Use a graphing utility to find the first real zero of the function. Confirm using synthetic division. 4) Find all remaining zeros of the function. ...
... 3) Use a graphing utility to find the first real zero of the function. Confirm using synthetic division. 4) Find all remaining zeros of the function. ...