Verifying Subring: Check Closure with ℂ & \alpha

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In summary, the conversation discusses verifying that the set R, defined as {x + yα | x,y ∈ ℤ} where α = (1+ √-3)/2 ∈ ℂ, is a subring of ℂ. The issue at hand is that when multiplying two elements of R to check closure, the result is the negative complex conjugate of α, which may indicate a mistake in the calculation.
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blahblah8724
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For [itex]\alpha = (1+ \sqrt{-3})/2 \in ℂ[/itex] and [itex]R = \{ x +y\alpha \, | \, x,y \in Z \}[/itex].

How would you verify that R is a subring of ℂ? Everytime I multiply two 'elements' of R to check closure I get the negative complex conjugate of [itex]\alpha[/itex], I think I'm doing something wrong...

Thanks!
 
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  • #2
blahblah8724 said:
For [itex]\alpha = (1+ \sqrt{-3})/2 \in ℂ[/itex] and [itex]R = \{ x +y\alpha \, | \, x,y \in Z \}[/itex].

How would you verify that R is a subring of ℂ? Everytime I multiply two 'elements' of R to check closure I get the negative complex conjugate of [itex]\alpha[/itex], I think I'm doing something wrong...

Thanks!

Let's see...

The negative complex conjugate of ##\alpha## is:
$$-\overline{\alpha} = - \frac 1 2 (1 - \sqrt{-3}) = \frac 1 2 (1 + \sqrt{-3}) - 1 = \alpha - 1$$

What exactly is the problem?
 

Related to Verifying Subring: Check Closure with ℂ & \alpha

1. What is a subring and how is it verified?

A subring is a subset of a larger ring that also forms a ring when using the same operations. To verify a subring, we need to check that it satisfies the three properties of a ring: closure, associativity, and distributivity. In this particular case, we are checking the closure property using the complex numbers and a given element alpha.

2. Can any subset of a ring be considered a subring?

No, not every subset of a ring is a subring. For a subset to be considered a subring, it must satisfy the three properties of a ring: closure, associativity, and distributivity. If these properties are not satisfied, then the subset cannot be considered a subring.

3. How do we check closure when verifying a subring?

To check closure, we need to make sure that when we perform the ring operations (addition, subtraction, and multiplication) on any two elements in the subset, the result is also in the subset. In this case, we are using the complex numbers and a given element alpha to check closure.

4. Can we use any element from the ring to check closure when verifying a subring?

Yes, we can use any element from the ring to check closure when verifying a subring. As long as the subset satisfies the closure property for that specific element, it can be considered a subring. However, it is important to note that using different elements may result in different subsets being considered subrings.

5. What is the significance of verifying a subring?

Verifying a subring is important because it allows us to identify subsets of a larger ring that also have the structure of a ring. This can help in understanding and solving problems in abstract algebra, as well as in applications such as cryptography and coding theory.

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