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With a few colleagues, we're trying to design an (intermediate) algebra course (US terminology) where we stress the interplay between algebra and geometry. The algebraic topics we would like to cover are (1) linear equation in two variables, (2) quadratic equations in two variables, (3) polynomials in one variable, (4) rational functions in one variable (though we're not sure we want to introduce functions), (5) radicals.

For (1) and (2) there are obvious geometric counterparts: lines and conic sections.

Question: Are there natural geometric counterparts for (3), (4) and (5)? Are there elementary geometric constructions that naturally lead to these algebraic objects?

Side question: Are there (affordable) textbooks or lecture notes out there which have this kind of approach?

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    Yes, if you're willing to work with _complex_ numbers there is a beautiful geometry of rational functions.2011-02-08
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    @Qiaochu Yuan: Could you elaborate a little? Thanks.2011-02-09
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    see, for example, the visualizations at http://en.wikipedia.org/wiki/M%C3%B6bius_transformation . Of course there is lot to be said about the interplay of algebra and geometry but unfortunately I don't know how much of it can be said at this level. If you really are serious about this goal don't think it's natural to restrict the choice of algebraic topics. I think a careful introduction to complex numbers alone would fill up a great course but again I have no idea if it could be made level-appropriate.2011-02-09
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    You want polynomials without defining functions???2011-04-10

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