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Computing the Second Exterior Power of Certain Ideals in (\mathbb$\mathbb{Z}[\sqrt{-5}])]$ and (\mathbb$\mathbb{Z}[\sqrt{5}])]$ as Modules

I'm working on a problem involving the computation of the second exterior power of certain ideals within the rings ( R1 = \mathbb{Z}[\sqrt{-5}]$R_1 = \mathbb{Z}[\sqrt{-5}]$ and R2 = \mathbb{Z}[\sqrt{5}])$R_2 = \mathbb{Z}[\sqrt{5}]$. The problem is as follows:

Let an ideal ( I = (2, 1 + \sqrt{-5}) )$I = (2, 1 + \sqrt{-5})$ as an ( R1 )$R_1$-module, I need to verify that the second exterior power ( \Lambda^2 I = 0 )$\bigwedge^2 I = 0$.

Let another ideal ( J = (2, 1 + \sqrt{5}) )$J = (2, 1 + \sqrt{5})$ as an ( R2 )$R_2$-module. However,I I need to prove that ( \Lambda^2 J \neq 0 )$\bigwedge^2 J \neq 0$.

Any insights on these computations or the underlying structure of the exterior power in this setting would be greatly appreciated.

Computing the Second Exterior Power of Certain Ideals in (\mathbb{Z}[\sqrt{-5}]) and (\mathbb{Z}[\sqrt{5}]) as Modules

I'm working on a problem involving the computation of the second exterior power of certain ideals within the rings ( R1 = \mathbb{Z}[\sqrt{-5}] and R2 = \mathbb{Z}[\sqrt{5}]). The problem is as follows:

Let an ideal ( I = (2, 1 + \sqrt{-5}) ) as an ( R1 )-module, I need to verify that the second exterior power ( \Lambda^2 I = 0 ).

Let another ideal ( J = (2, 1 + \sqrt{5}) ) as an ( R2 )-module. However,I need to prove that ( \Lambda^2 J \neq 0 ).

Any insights on these computations or the underlying structure of the exterior power in this setting would be greatly appreciated.

Computing the Second Exterior Power of Certain Ideals in $\mathbb{Z}[\sqrt{-5}]$ and $\mathbb{Z}[\sqrt{5}]$ as Modules

I'm working on a problem involving the computation of the second exterior power of certain ideals within the rings $R_1 = \mathbb{Z}[\sqrt{-5}]$ and $R_2 = \mathbb{Z}[\sqrt{5}]$. The problem is as follows:

Let an ideal $I = (2, 1 + \sqrt{-5})$ as an $R_1$-module, I need to verify that the second exterior power $\bigwedge^2 I = 0$.

Let another ideal $J = (2, 1 + \sqrt{5})$ as an $R_2$-module. However, I need to prove that $\bigwedge^2 J \neq 0$.

Any insights on these computations or the underlying structure of the exterior power in this setting would be greatly appreciated.

deleted 22 characters in body; edited title
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Haze
Haze
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Computing the Second Exterior Power of Certain Ideals in (\mathbb{Z}[\sqrt{-5}]) and (\mathbb{Z}[\sqrt{5}]) as Modules

I'm working on a problem involving the computation of the second exterior power of certain ideals within the ringrings ( RR1 = \mathbb{Z}[\sqrt{-5}] and R2 = \mathbb{Z}[\sqrt{5}]). The problem is as follows:

Let an ideal ( I = (2, 1 + \sqrt{-5}) ) be an ideal in ( R ). When regarded as an ( RR1 )-module, I need to verify that the second exterior power ( \Lambda^2 I = 0 ).

Let another ideal ( J = (2, 1 + \sqrt{5}) ) be another ideal in ( R ). However, when regarded as an ( RR2 )-module. However, II need to prove that ( \Lambda^2 J \neq 0 ).

Any insights on these computations or the underlying structure of the exterior power in this setting would be greatly appreciated.

Computing the Second Exterior Power of Certain Ideals in (\mathbb{Z}[\sqrt{-5}]) as Modules

I'm working on a problem involving the computation of the second exterior power of certain ideals within the ring ( R = \mathbb{Z}[\sqrt{-5}] ). The problem is as follows:

Let ( I = (2, 1 + \sqrt{-5}) ) be an ideal in ( R ). When regarded as an ( R )-module, I need to verify that the second exterior power ( \Lambda^2 I = 0 ).

Let ( J = (2, 1 + \sqrt{5}) ) be another ideal in ( R ). However, when regarded as an ( R )-module, I need to prove that ( \Lambda^2 J \neq 0 ).

Any insights on these computations or the underlying structure of the exterior power in this setting would be greatly appreciated.

Computing the Second Exterior Power of Certain Ideals in (\mathbb{Z}[\sqrt{-5}]) and (\mathbb{Z}[\sqrt{5}]) as Modules

I'm working on a problem involving the computation of the second exterior power of certain ideals within the rings ( R1 = \mathbb{Z}[\sqrt{-5}] and R2 = \mathbb{Z}[\sqrt{5}]). The problem is as follows:

Let an ideal ( I = (2, 1 + \sqrt{-5}) ) as an ( R1 )-module, I need to verify that the second exterior power ( \Lambda^2 I = 0 ).

Let another ideal ( J = (2, 1 + \sqrt{5}) ) as an ( R2 )-module. However,I need to prove that ( \Lambda^2 J \neq 0 ).

Any insights on these computations or the underlying structure of the exterior power in this setting would be greatly appreciated.

Source Link
Haze
Haze
  • 93
  • 5

Computing the Second Exterior Power of Certain Ideals in (\mathbb{Z}[\sqrt{-5}]) as Modules

I'm working on a problem involving the computation of the second exterior power of certain ideals within the ring ( R = \mathbb{Z}[\sqrt{-5}] ). The problem is as follows:

Let ( I = (2, 1 + \sqrt{-5}) ) be an ideal in ( R ). When regarded as an ( R )-module, I need to verify that the second exterior power ( \Lambda^2 I = 0 ).

Let ( J = (2, 1 + \sqrt{5}) ) be another ideal in ( R ). However, when regarded as an ( R )-module, I need to prove that ( \Lambda^2 J \neq 0 ).

Any insights on these computations or the underlying structure of the exterior power in this setting would be greatly appreciated.