By Michal Krizek, Florian Luca, Lawrence Somer, A. Solcova
The pioneering paintings of French mathematician Pierre de Fermat has attracted the eye of mathematicians for over 350 years. This booklet used to be written in honor of the four-hundredth anniversary of his start, supplying readers with an summary of the numerous houses of Fermat numbers and demonstrating their functions in components equivalent to quantity thought, chance conception, geometry, and sign processing. This publication introduces a common mathematical viewers to uncomplicated mathematical rules and algebraic equipment attached with the Fermat numbers.
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Extra resources for 17 Lectures on Fermat Numbers: From Number Theory to Geometry
Moreover, it is also obvious that setting ϕ (m, n) = α(m, n) is the only possible choice such that α = ϕ ◦ τ. Hence the universal property is satisfied, and T together with τ is indeed a tensor product. 6 (Tensor products). Let M and N be R-modules. 5 there is a unique tensor product of M and N over R up to isomorphism, i. e. 2 (b). We write T as M ⊗R N (or simply M ⊗ N if the base ring is understood), and τ(m, n) as m ⊗ n. The element m ⊗ n ∈ M ⊗ N is often called the tensor product of m and n.
We claim that R[x, y] ∼ = R[x] ⊗R R[y] as R-algebras, i. e. that polynomial rings in several variables can be thought of as tensor products of polynomial rings in one variable. In fact, there are R-module homomorphisms ϕ : R[x] ⊗R R[y] → R[x, y], f ⊗ g → f g (by the universal property of the tensor product) and ψ : R[x, y] → R[x] ⊗ R[y], ∑ ai, j xi y j → ∑ ai, j xi ⊗ y j . i, j i, j As (ψ ◦ ϕ)(xi ⊗ y j ) = ψ(xi y j ) = xi ⊗ y j and (ϕ ◦ ψ)(xi y j ) = ϕ(xi ⊗ y j ) = xi y j for all i, j ∈ N and these elements xi ⊗ y j and xi y j generate R[x] ⊗R R[y] and R[x, y] as an R-module, respectively, we see that ϕ and ψ are inverse to each other.
N − 1 by the exactness of the sequence. Plugging this into the above formula, the result follows. Of course, knowing the length of a module does not mean that one knows the module up to isomorphism (as we have seen e. g. 21). So let us now address the question to what extent a module in an exact sequence can be completely recovered from the other parts of the sequence. For simplicity, let us restrict to short exact sequences. 11 (Recovering modules from an exact sequence). Consider an exact sequence ϕ ψ 0 −→ M −→ N −→ P −→ 0 of R-modules, and let us ask whether we can determine one of the three modules if we know the rest of the sequence.