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Teoria multiplicativa a idealelor

 
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dede
Euclid


Joined: 16 Oct 2007
Posts: 37

PostPosted: Fri Jul 29, 2011 6:01 pm    Post subject: Teoria multiplicativa a idealelor Reply with quote

Profesor: Tiberiu Dumitrescu

1) Aratati ca in \mathbb{Z}[\sqrt{-5}] elementele 3, 7, 1 \pm 2 \sqrt{-5} sunt ireductibile si deduceti ca \mathbb{Z}[\sqrt{-5}] nu e factorial.

2) Aratati ca extinderea  \mathbb{Z} \subseteq \mathbb{Z}[\sqrt{2+ \sqrt{5+ \sqrt{7}}},i] este intreaga.

3) Fie D un domeniu noetherian si I un ideal nenul. Aratati ca I este t-inversabil daca si numai daca  I_v : I = D .

4) Aratati ca un domeniu D este Prufer daca si numai daca oricare ar fi  a,b,c \in D \setminus \{0\} , avem  (aD +bD+cD)(abD \cap bcD \cap acD)=abcD.

5) Este functia  I \rightarrow \sqrt{I} o star operatie pe Z[i]?

6) Fie  D = Z[\sqrt{-34}] si G grupul Picard al lui D. Presupunem cunoscut faptul ca ordinul lui G este 4. In D, gasiti 2 descompuneri ale lui 625 in produs de atomi, Este G ciclic?
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