- #1
Brimley
- 77
- 0
Hello PhysicsForums!
I have been reading up on congruence classes and working out some examples. I came across one example that I seem to struggle understanding.
I've solved for [itex]\lambda[/itex] and I know that [itex]\lambda = (3+\sqrt{-3})/2[/itex] [itex]\in[/itex] [itex]Q[\sqrt{-3}][/itex]. I also know that [itex]\lambda[/itex] is a prime in [itex]Q[\sqrt{-3}][/itex].
From here, I would like to prove that iff [itex]\lambda[/itex] divides [itex]a[/itex] for some rational integer [itex]a[/itex] in [itex]Z[/itex], it can be proven that 3 divides [itex]a[/itex].
Can this is done? If so, could someone show me?
Lastly (or as a second part to this), what are the congruence classes [itex] (mod (3+\sqrt{3})/2) [/itex] in [itex] Q[\sqrt{-3}] [/itex] ?
I really appreciate the help on this everyone!
*Note: I intentionally put [itex] (mod (3+\sqrt{3})/2) [/itex] with the [itex] \sqrt{3} [/itex], so it should not be negative for this part.
I have been reading up on congruence classes and working out some examples. I came across one example that I seem to struggle understanding.
I've solved for [itex]\lambda[/itex] and I know that [itex]\lambda = (3+\sqrt{-3})/2[/itex] [itex]\in[/itex] [itex]Q[\sqrt{-3}][/itex]. I also know that [itex]\lambda[/itex] is a prime in [itex]Q[\sqrt{-3}][/itex].
From here, I would like to prove that iff [itex]\lambda[/itex] divides [itex]a[/itex] for some rational integer [itex]a[/itex] in [itex]Z[/itex], it can be proven that 3 divides [itex]a[/itex].
Can this is done? If so, could someone show me?
Lastly (or as a second part to this), what are the congruence classes [itex] (mod (3+\sqrt{3})/2) [/itex] in [itex] Q[\sqrt{-3}] [/itex] ?
I really appreciate the help on this everyone!
*Note: I intentionally put [itex] (mod (3+\sqrt{3})/2) [/itex] with the [itex] \sqrt{3} [/itex], so it should not be negative for this part.