# Download e-book for iPad: Abstract Algebra with Applications by Karlheinz Spindler

By Karlheinz Spindler

A entire presentation of summary algebra and an in-depth therapy of the purposes of algebraic suggestions and the connection of algebra to different disciplines, akin to quantity conception, combinatorics, geometry, topology, differential equations, and Markov chains.

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Let x ∈ P and y ∈ O. Then x y ∈ O∗ for otherwise for some z ∈ O we would have x yz = 1 and x would also have an inverse in O. This proves that x y ∈ P. Let then x, y ∈ P \ {0}. Then either y/x ∈ O or x/y ∈ O, and we can suppose by symmetry that the former option holds. Hence x − y = x(1 − (y/x)) = xt, with t := 1 − (y/x) ∈ O. Therefore, in view of what we have just seen, we have x − y ∈ P. These veriUcations show that P is an ideal. Since P is by deUnition the set of non invertible elements of O, we deduce that P is the unique maximal ideal of O, which is therefore a local ring.

4. We have |x| ∈ {cm : m ∈ Z} ∪ {0} if x ∈ K , and |x| = 0 if and only if x = 0. Also, we have |x y| = |x||y| for x, y ∈ K and, for x1 , . . , xn ∈ K , it holds that |x1 + . . ,n |xi | with equality if the maximum is attained at a single index. These properties say in particular that the absolute value | · | induces a distance d(· , ·) on K 2 , deUned by d(x, y) := |x − y|. 5) Such a distance in turn gives a topology on the Ueld K , which becomes a metric space. It is an easy matter to verify that the Ueld operations are continuous, which makes K a topological Ueld.

The numerator of the rational function F(ϕ1 , . . , ϕm ) thus corresponds to a relation among the y j . 15 Then, taking into account what we have just observed on the relations among the xi and the yi , it is not difUcult to verify that the inclusion K ⊂ L corresponds to a nonconstant rational map ϕ = (ϕ1 , . . , ϕm ) from C L to C K ; namely, this map sends a point (a1 , . . , an ) ∈ C L in (ϕ1 (a1 , . . , an ), . . , ϕm (a1 , . . , an )) ∈ C K . It is important to note that such ϕ is not quite a function deUned on the whole curve C L : it shall be generally deUned only on the points P ∈ C L such that ϕ1 , .