Why does sqrt(-2) divide x in Z[sqrt(-2)] if it divides x+sqrt(-2)?

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Discussion Overview

The discussion revolves around the divisibility of elements in the ring Z[sqrt(-2)], specifically addressing the condition under which sqrt(-2) divides an integer x given that it divides the sum x + sqrt(-2). The scope includes mathematical reasoning and properties of divisibility within this algebraic structure.

Discussion Character

  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants propose that if sqrt(-2) divides x + sqrt(-2), then it should also divide x, suggesting a property of divisibility in sums.
  • Others challenge this reasoning by providing a counterexample, indicating that a number can divide a sum without dividing each individual term.
  • A participant seeks clarification on the original question, asking for a more detailed explanation of the divisibility condition in the context of the ring Z[sqrt(-2)].
  • Another participant agrees with the need for clarification and reiterates the original question regarding the proof of divisibility.
  • One participant references a specific property of divisibility, stating that if x divides y + z and x divides y, then x must divide z, providing a proof for this claim.
  • Another participant expresses agreement with this property, indicating a shared understanding of the mathematical principle discussed.

Areas of Agreement / Disagreement

Participants express differing views on the validity of the initial claim regarding divisibility. While some support the idea that sqrt(-2) dividing x + sqrt(-2) implies it divides x, others dispute this by citing counterexamples. The discussion remains unresolved with multiple competing views present.

Contextual Notes

There are limitations in the assumptions made about the properties of divisibility in the context of the ring Z[sqrt(-2)], and the discussion does not fully resolve the implications of these properties.

Poirot1
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Why does it follow that if x is an integer such that sqrt(-2)|x+sqrt(-2) in Z[sqrt(-2)], then
sqrt(-2)|x?
 
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Poirot said:
Why does it follow that if x is an integer such that sqrt(-2)|x+sqrt(-2) in Z[sqrt(-2)], then
sqrt(-2)|x?

Wouldn't it be because in order for ANY number to evenly divide into a sum, it must divide into each term in the sum evenly?
 
Poirot said:
Why does it follow that if x is an integer such that sqrt(-2)|x+sqrt(-2) in Z[sqrt(-2)], then
sqrt(-2)|x?

Hi Poirot, :)

Can you please elaborate more about your question. Do you mean,

\[\frac{x}{\sqrt{-2}}+\sqrt{-2}\in \mathbb{Z}[\sqrt{-2}]=\{a+b\sqrt{-2} ; a\in \mathbb Z,b\in \mathbb Z \}\]

then prove that, \(\sqrt{-2}|x\) ?
 
Sudharaka said:
Hi Poirot, :)

Can you please elaborate more about your question. Do you mean,

\[\frac{x}{\sqrt{-2}}+\sqrt{-2}\in \mathbb{Z}[\sqrt{-2}]=\{a+b\sqrt{-2} ; a\in \mathbb Z,b\in \mathbb Z \}\]

then prove that, \(\sqrt{-2}|x\) ?

I believe the OP is asking that if [math]\displaystyle \sqrt{-2}[/math] evenly divides [math]\displaystyle x + \sqrt{-2}[/math], then to show that [math]\displaystyle \sqrt{-2}[/math] evenly divides x.
 
Of course. Thanks 'Prove it'.
 
Prove It said:
Wouldn't it be because in order for ANY number to evenly divide into a sum, it must divide into each term in the sum evenly?

I don't follow. We have $5|2+3$, yet five divides neither of the summands.
Are you referencing a certain property of this ring?
 
The Chaz said:
I don't follow. We have $5|2+3$, yet five divides neither of the summands.
Are you referencing a certain property of this ring?

To be more precise, if x|y+z and x|y, then x|z.
Pf. xm=y+z and xn=y for some m,n. Thus z=x(m-n) so that x|z
 
Poirot said:

To be more precise, if x|y+z and x|y, then x|z.
Pf. xm=y+z and xn=y for some m,n. Thus z=x(m-n) so that x|z

Agreed!
 

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