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I am reading Joseph J. Rotman's book: A First Course in Abstract Algebra with Applications (Third Edition) ...
I am currently focused on Section 3.5 From Numbers to Polynomials ...
I need help with Example 3.76 ... ... the example concerns Euclidean rings and their defining characteristics so I am including the definition of a Euclidean ring in the relevant text shown below ... ...
The relevant text from Rotman's book is as follows:https://www.physicsforums.com/attachments/4649
View attachment 4648
I am trying to understand Example 3.76 which indicates that every field is a Euclidean ring ... ...
I can see that point (i) of the definition is satisfied with $$\partial$$ set identically to zero ...... BUT ... I fail to understand what Rotman is saying about how point (ii) is satisfied ... ...
In order for (ii) to be satisfied, for every $$g \in R$$ and every $$f \in R^{\times}$$ we have to find $$q, r \in R$$ such that:
$$g = qf + r$$ ... ... ... (*)
... BUT ...
Rotman says to set $$q = f^{-1}$$ and $$r = 0$$
but if we do this (*) above becomes
$$g = f f^{-1} + 0 = 1$$ ...
but $$g$$ may be any element of $$R$$ ... ?Can someone please explain what is going on ... that is, what Rotman means in this example ...
Hope someone can help ...
Peter
I am currently focused on Section 3.5 From Numbers to Polynomials ...
I need help with Example 3.76 ... ... the example concerns Euclidean rings and their defining characteristics so I am including the definition of a Euclidean ring in the relevant text shown below ... ...
The relevant text from Rotman's book is as follows:https://www.physicsforums.com/attachments/4649
View attachment 4648
I am trying to understand Example 3.76 which indicates that every field is a Euclidean ring ... ...
I can see that point (i) of the definition is satisfied with $$\partial$$ set identically to zero ...... BUT ... I fail to understand what Rotman is saying about how point (ii) is satisfied ... ...
In order for (ii) to be satisfied, for every $$g \in R$$ and every $$f \in R^{\times}$$ we have to find $$q, r \in R$$ such that:
$$g = qf + r$$ ... ... ... (*)
... BUT ...
Rotman says to set $$q = f^{-1}$$ and $$r = 0$$
but if we do this (*) above becomes
$$g = f f^{-1} + 0 = 1$$ ...
but $$g$$ may be any element of $$R$$ ... ?Can someone please explain what is going on ... that is, what Rotman means in this example ...
Hope someone can help ...
Peter
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