Find all elements a+b(-3)^0.5 in Z[-3^0.5] such that |a+b(-3)^0.5|^2=1

**mbmstudent** · March 8th 2011, 05:22 AM

Hi,

I have the following problem:

"Find all elements $a+b(-3)^{0.5}$ in $Z[-3^{0.5}]$ such that $|a+b(-3)^{0.5}|^2=1$ "

And I think I have the answer, but I don't know if it is right since it says "elements" and not "element."

Well, I know that $|a+b(-3)^{0.5}|^2=a+b(-3)^{0.5}$ (obvious), and that if $a+b(-3)^{0.5}$ is a unit, then $|a+b(-3)^{0.5}|^2=1$ ... is that correct?

Thanks!

**FernandoRevilla** · March 8th 2011, 05:34 AM

$(a+b\;\sqrt{-3})^2=a^2-3b^2+2ab\;\sqrt{-3}=1\Leftrightarrow a^2-3b^2=1\;\wedge\;2ab=0$

Now, analyze the cases $a=0$ and $b=0$ .

**tonio** · March 8th 2011, 05:34 AM

Originally Posted by mbmstudent

Hi,

I have the following problem:

"Find all elements $a+b(-3)^{0.5}$ in $Z[-3^{0.5}]$ such that $|a+b(-3)^{0.5}|^2=1$ "

And I think I have the answer, but I don't know if it is right since it says "elements" and not "element."

Well, I know that $|a+b(-3)^{0.5}|^2=a+b(-3)^{0.5}$ (obvious)

Not ony not obvious but even wrong: $|a+b\sqrt{-3}|^2=a^2+3b^2$ , since we're talking here about

the module (=absolute value) of a complex number

, and that if $a+b(-3)^{0.5}$ is a unit, then $|a+b(-3)^{0.5}|^2=1$ ... is that correct?

Thanks!

In fact, $a^2+3b^2=1\Longleftrightarrow a=\pm 1$ , so we have two different units here.

Tonio

**tonio** · March 8th 2011, 05:34 AM

Originally Posted by FernandoRevilla

$(a+b\;\sqrt{-3})^2=a^2-3b^2+2ab\;\sqrt{-3}=1\Leftrightarrow a^2-3b^2=1\;\wedge\;2ab=0$

Now, analyze the cases $a=0$ and $b=0$ .

I think you missed the absolute value (module) symbol the OP wrote in his mesage.

Tonio

**FernandoRevilla** · March 8th 2011, 05:53 AM

Originally Posted by tonio

I think you missed the absolute value (module) symbol the OP wrote in his mesage.

Of course.

Thread: Find all elements a+b(-3)^0.5 in Z[-3^0.5] such that |a+b(-3)^0.5|^2=1

LinkBack

Thread Tools

Display

Find all elements a+b(-3)^0.5 in Z[-3^0.5] such that |a+b(-3)^0.5|^2=1

Similar Math Help Forum Discussions

Keeping latex elements in alignment with non-latex elements

Suppose the set A has m elements and the set B has n elements. there are 2^mn relatio

Find the number of elements in the intersection of 3 sets

find the elements

find the elements of S_4

Search Tags