Timeline for Computing the kernel of some Artin-Map
Current License: CC BY-SA 4.0
10 events
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| Apr 1, 2020 at 11:25 | comment | added | Julian | Then in the next step I want to show that the kernel of $\varphi_{L/K}$ is exactly $P_K$. Because I know excatly which prime ideals maps to $1 \in Gal(L/K$, I can see that every product of principal prime ideals will be mapped to the $1$. Every Product of even many non-principal prime ideals will be mapped to the $1$, too. Because the class number of $K$ is $2$ every product of non-principal ideals must be a principal ideal. Can I conclude from this that every non-principal(resp. principal) Ideal is the product of some principal ideals and uneven(resp. even) man non-principal ideals? | |
| Apr 1, 2020 at 5:50 | comment | added | Franz Lemmermeyer | a) No: the product of two nonprincipal prime ideals is principal. b) 2 is inert in one subfields, 5 splits in another. | |
| Mar 31, 2020 at 20:18 | comment | added | Julian | Ah, now I see: $p\mathcal{O}_K =(x+y\sqrt{-5})(x-y\sqrt{-5}) = (x^2 +5y^2)$ and this is exactly the case iff $p \equiv 1,9\ mod\ 20$. And for $p \equiv 11,13,17,19\ mod\ 20$ $p\mathcal{O}_K$ remains prime and is principal. Now I only need to compute $\varphi(2\mathcal{O}_K)$ and $\varphi(5\mathcal{O}_K)$. Two another questions: Is $P_K$ generated by the principal prime ideals and how can I see in which ways $2\mathcal{O}_K,5\mathcal{O}_K$ splits in $L$? And thank you for your help. | |
| Mar 31, 2020 at 19:55 | comment | added | Franz Lemmermeyer | Yes. Principal ideals $(x,+y\sqrt{-5})$ have norm $p = x^2 + 5y^2$. The nonprincipal primes lie in the class of the prime ideal above $2$ . . . | |
| Mar 31, 2020 at 19:31 | history | edited | Julian | CC BY-SA 4.0 |
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| Mar 31, 2020 at 19:15 | comment | added | Julian | When you talk about norm, do you mean $\mathcal{N}(\mathfrak{p}) = |\mathcal{O}_K/\mathfrak{p}|$ for a prime ideal $\mathfrak{p}$ in $K$? | |
| Mar 31, 2020 at 18:58 | history | edited | Julian | CC BY-SA 4.0 |
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| Mar 31, 2020 at 18:42 | comment | added | Franz Lemmermeyer | Everything boils down to showing that prime ideals with norm $\equiv 1 \bmod 4$ are principal. This is elementary. You have to show that primes $p$ that split satisfy $p = x^2 + 5y^2$ if $p \equiv 1 \bmod 4$ and $2p = x^2 + 5y^2$ if $p \equiv 3 \bmod 4$. | |
| Mar 31, 2020 at 18:40 | history | edited | Franz Lemmermeyer | CC BY-SA 4.0 |
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| Mar 31, 2020 at 2:48 | history | asked | Julian | CC BY-SA 4.0 |