Abstract
In this chapter, we will explore the correspondence between ideals and varieties. In §§1 and 2, we will prove the Nullstellensatz, a celebrated theorem which identifies exactly which ideals correspond to varieties. This will allow us to construct a “dictionary” between geometry and algebra, whereby any statement about varieties can be translated into a statement about ideals (and conversely). We will pursue this theme in §§3 and 4, where we will define a number of natural algebraic operations on ideals and study their geometric analogues.
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© 2007 Springer Science+Business Media, LLC
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Cox, D., Little, J., O’Shea, D. (2007). The Algebra–Geometry Dictionary. In: Ideals, Varieties, and Algorithms. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-35651-8_4
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DOI: https://doi.org/10.1007/978-0-387-35651-8_4
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-35650-1
Online ISBN: 978-0-387-35651-8
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