Determine whether this is a subfield of R

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Homework Help Overview

The discussion revolves around determining whether the set S = {r + s√2 | r, s ∈ Q} is a subfield of the real numbers R. The subject area pertains to field theory within abstract algebra.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants explore the closure of set S under field axioms, particularly questioning whether the division of elements in S remains within S. There is a focus on the implications of the structure of S and its elements.

Discussion Status

The discussion is ongoing, with some participants providing guidance on how to approach the problem, particularly regarding the closure properties of S. Multiple interpretations of the requirements for S to be a subfield are being explored.

Contextual Notes

There are indications of confusion regarding the terminology used in the problem statement, which may affect the clarity of the discussion.

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Let Q denote the field of rational numbers and R denote the field of real numbers. Determine whether or not set S = {r + s√2 | r, s € Q} is a subfield of R.
 
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Belong to quotient of what? That's not very grammatical.
 
Let Q denote the field of rational numbers and R denote the field of real numbers. Determine whether or not set S = {r + s√2 | r, s € Q} is a subfield of R.
 
That's better. Just show S is closed under the field axioms. The big question is if a and b are in S is a/b in S? If b=r+sqrt(2)s it's useful to multiply the numerator and denominator of a/b by (r-sqrt(2)s).
 

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