Smallest subfields containing Z and Z[sqrt2]

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SUMMARY

The smallest subfields containing Z[i] and Z[√2] are Q[i] and Q[√2], respectively. The combined subfield that encompasses both Z[i] and Z[√2] is Q[i, √2]. This conclusion is derived from the definitions of the sets, where Z[i] consists of all complex numbers of the form a + ib (with a, b in Z) and Z[√2] consists of all numbers of the form a + b√2 (with a, b in Z). The solution confirms that Q[i, √2] is indeed the smallest field containing both Z[i] and Z[√2].

PREREQUISITES
  • Understanding of complex numbers and their representation, specifically Z[i]
  • Familiarity with field theory and subfields
  • Knowledge of real numbers and their extensions, particularly Z[√2]
  • Basic algebra involving polynomial expressions and roots
NEXT STEPS
  • Study field extensions in algebra, focusing on Q[i] and Q[√2]
  • Explore the properties of Q[i, √2] as a field
  • Learn about the construction of subfields from integral domains
  • Investigate other examples of smallest subfields containing various algebraic integers
USEFUL FOR

Mathematics students, particularly those studying abstract algebra and field theory, as well as educators looking for examples of field extensions and subfields.

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Smallest subfields containing Z and Z[sqrt2]

Homework Statement


What are the smallest subfields of R containing Z and Z[\sqrt{2}]?


Homework Equations


Z= {a+ib|a,b\in Z}
Z[\sqrt{2}]={a+b\sqrt{2}|a,b\in Z}

The Attempt at a Solution


Z\subsetQ and Z[\sqrt{2}]\subsetQ[\sqrt{2}]
so to get the subfield that contains both of them, would it be Q[i,\sqrt{2}]?
 
Physics news on Phys.org


Yes.
 

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