کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
5771671 | 1630421 | 2018 | 21 صفحه PDF | دانلود رایگان |
عنوان انگلیسی مقاله ISI
The number of ideals of Z[x] containing x(x â α)(x â β) with given index
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موضوعات مرتبط
مهندسی و علوم پایه
ریاضیات
اعداد جبر و تئوری
پیش نمایش صفحه اول مقاله
چکیده انگلیسی
It is well-known that a connected regular graph is strongly-regular if and only if its adjacency matrix has exactly three eigenvalues. Let B denote an integral square matrix and ãBã denote the subring of the full matrix ring generated by B. Then ãBã is a free Z-module of finite rank, which guarantees that there are only finitely many ideals of ãBã with given finite index. Thus, the formal Dirichlet series ζãBã(s)=ânâ¥1annâs is well-defined where an is the number of ideals of ãBã with index n. In this article we aim to find an explicit form of ζãBã(s) when B has exactly three eigenvalues all of which are integral, e.g., the adjacency matrix of a strongly-regular graph which is not a conference graph with a non-squared number of vertices. By isomorphism theorem for rings, ãBã is isomorphic to Z[x]/m(x)Z[x] where m(x) is the minimal polynomial of B over Q, and Z[x]/m(x)Z[x] is isomorphic to Z[x]/m(x+γ)Z[x] for each γâZ. Thus, the problem is reduced to counting the number of ideals of Z[x]/x(xâα)(xâβ)Z[x] with given finite index where 0,α and β are distinct integers.
ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Algebra - Volume 493, 1 January 2018, Pages 36-56
Journal: Journal of Algebra - Volume 493, 1 January 2018, Pages 36-56
نویسندگان
Mitsugu Hirasaka, Semin Oh,