Timeline for Does the unit index divide the degree of an extension of number fields?
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| when toggle format | what | by | license | comment | |
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| May 19, 2017 at 16:39 | comment | added | Alex B. | ...is $3$. For the details see arxiv.org/abs/0904.2416, particularly the table in Example 6.4, which lists the possible $\mathbb{Z}[S_3]$-module structure of $U_L$ and the corresponding value for a certain quotient of class numbers. | |
| May 19, 2017 at 16:37 | comment | added | Alex B. | @A.Maarefparvar: In the special case that $L/\mathbb{Q}$ is a real $S_3$-extension, my last paragraph applies to the relative extension, but you can also sometimes exploit the extra structure. Indeed, if you know the $\mathbb{Z}_3[S_3]$-module structure of $U_L$, then you also know the $\mathbb{Z}_3[C_3]$-module structure. For the $S_3$-structure, there are only finitely many possibilities, and they can sometimes be read off from the class numbers of the intermediate extensions. In particular, if a certain quotient of class numbers is equal to $1/9$, then you will know that the norm index... | |
| May 19, 2017 at 12:10 | comment | added | A. Maarefparvar | But in case L is real, when the fundamental unit of K is not norm of any unit of L, I don't have any idea!! | |
| May 19, 2017 at 12:09 | vote | accept | A. Maarefparvar | ||
| May 19, 2017 at 12:08 | comment | added | A. Maarefparvar | I am so grateful for very useful your comments. But I guess for our subject , the statement is true. Indeed, I consider a special case that L is an S_3 -extension of Q and K it's unique quadratic subfield . If L is maginary, and K has no a primitive third root of unity, hence the index is 1, otherwise 1 or 3, right?? | |
| May 18, 2017 at 18:11 | history | edited | Alex B. | CC BY-SA 3.0 |
added 746 characters in body
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| May 18, 2017 at 17:03 | history | answered | Alex B. | CC BY-SA 3.0 |