Algebraic Dependence of (x,z): If (x,y,z) & (x,y) Dependent

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SUMMARY

The discussion centers on the algebraic dependence of the ordered pairs and triples, specifically whether the algebraic dependence of (x,y,z) and (x,y) implies the algebraic dependence of (x,z). It is established that any subset of an algebraically independent set remains algebraically independent. Additionally, participants seek resources for understanding transcendence degrees and definitions related to algebraic dependence.

PREREQUISITES
  • Understanding of algebraic independence and dependence
  • Familiarity with ordered pairs and triples in algebra
  • Knowledge of transcendence degrees in algebraic geometry
  • Basic concepts of sets and subsets in mathematics
NEXT STEPS
  • Research the definitions and properties of algebraic independence
  • Explore the concept of transcendence degrees in algebraic structures
  • Study examples of algebraically dependent and independent sets
  • Investigate the implications of algebraic dependence in polynomial equations
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Mathematicians, algebra students, and educators seeking to deepen their understanding of algebraic structures and dependencies in mathematical contexts.

disregardthat
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If (x,y,z) and (x,y) is algebraically dependent, is then (x,z) algebraically dependent?

Does by the way anyone have a good web-source for information about transcendence degrees?
 
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Definitions needed. What are x,y,z? What does it mean for an ordered triple and and ordered pair to be algebraically dependent?
 
Any subset of an algebraically independent set is algebraically independent.
 

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