Determine whether this is a subfield of R

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In summary, the conversation discusses the determination of whether the set S, defined as {r + s√2 | r, s € Q}, is a subfield of the fields Q (rational numbers) and R (real numbers). The main question is whether the set S is closed under the field axioms, specifically if the quotient of two elements in S is also in S. To prove this, it is suggested to multiply the numerator and denominator of the quotient by (r - sqrt(2)s).
  • #1
dash
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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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  • #2
Belong to quotient of what? That's not very grammatical.
 
  • #3
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.
 
  • #4
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).
 

1. What is a subfield of R?

A subfield of R is a subset of the mathematical field of real numbers that satisfies the same properties as the larger field. In other words, it is a smaller set of numbers that still follows the same rules and operations as the entire set of real numbers.

2. How do you determine if something is a subfield of R?

To determine if something is a subfield of R, you must check if it satisfies the three properties of a field: closure, associativity, and inverses. If the subset follows these properties, it can be considered a subfield of R.

3. Can any set of numbers be a subfield of R?

No, not all sets of numbers can be considered a subfield of R. The set must satisfy the three properties of a field and must also be a subset of the real numbers. For example, the set of only even numbers would not be a subfield of R because it does not contain all real numbers.

4. Is R itself considered a subfield of R?

Yes, R is considered a subfield of R because it is a subset of itself and satisfies the three properties of a field.

5. What is the importance of identifying subfields of R?

Identifying subfields of R is important in understanding the properties and relationships between different sets of numbers. It also allows for more complex mathematical concepts to be broken down and analyzed in smaller, more manageable subsets.

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