Abstract | Ukratko, u prvom dijelu ovog rada promatra se algebarska struktura integralne domene i dokazuju osnovna svojstva. Navode se primjeri integralnih domena s posebnim naglaskom na skupove oblika \(\mathbb{Z} + \mathbb{Z} \sqrt{m}\) i \(\mathbb{Z} + \mathbb{Z} (\frac{1 + \sqrt{m}}{2})\) gdje je \(m\) kvadratno slobodan cijeli broj. Nadalje, u radu se definiraju asocirani, ireducibilni te prosti elementi integralne domene i neka njihova svojstva. Radi opsežnijeg istraživanja djeljivosti, rasprava je ograničena na posebnu klasu integralnih domena, a to su domene glavnih ideala. U drugom poglavlju definira se euklidska domena te promatraju skupovi koji tvore takvu strukturu posebno skupovi oblika \(\mathbb{Z} + \mathbb{Z} \sqrt{m}\) i \(\mathbb{Z} + \mathbb{Z} (\frac{1 + \sqrt{m}}{2})\). Zanimljivo je da ti skupovi tvore euklidsku domenu s obzirom na funkciju \(\phi_m(r+s \sqrt{m}) = |r^2-ms^2|\) ako i samo ako je \(m \in \{-1, -2, -3, -7, -11 \}\) za \(m<0\) i \(m \in \{ 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73 \}\) za \(m>0\). Dokazana svojstva omogućuju prikaz neparnog prostog broja \(p\) pomoću binarne kvadratne forme \((x^2 + y^2, x^2 + 2y^2, x^2 + xy + y^2, x^2 + xy + 2y^2\) i \(x^2 + xy + 3y^2)\). |
Abstract (english) | In the first part of this thesis we study the algebraic structure of integral domains and prove some basic properties of them. We give the examples of integral domains with the emphasis on sets of the form \(\mathbb{Z} + \mathbb{Z} \sqrt{m}\) and \(\mathbb{Z} + \mathbb{Z} (\frac{1 + \sqrt{m}}{2})\), where \(m\) is a squarefree integer. Moreover, associates, irreducible and prime elements of an integral domain with their properties are introduced. Also, for a deeper investigation of divisibility, we limit our discussion to a special class of integral domains - a principal ideal domain. In the second part, we discuss Euclidean domains especially among the sets \(\mathbb{Z} + \mathbb{Z} \sqrt{m}\) and \(\mathbb{Z} + \mathbb{Z} (\frac{1 + \sqrt{m}}{2})\). It is interesting that these sets are Euclidean domains with respect to the function \(\phi_m(r+s \sqrt{m}) = |r^2-ms^2|\) if and only if \(m \in \{-1, -2, -3, -7, -11 \}\) za \(m<0\) i \(m \in \{ 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73 \}\) for \(m<0\) and \(m \in \{ 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73 \}\) for \(m>0\). We apply these results to determine when an odd prime can be represented by a binary quadratic form \((x^2 + y^2, x^2 + 2y^2, x^2 + xy + y^2, x^2 + xy + 2y^2\) and \(x^2 + xy + 3y^2)\). |