In work expansion of 4 degrees of a ring of integers Z[α] = ({a0 +a1α + a2α2 + a3α3 | a0, a1, a2, a3 ∈ Z}, + , · ) is examined. The problem is solved: what should be integers x0, x1, x2, x3, that ξ = x0 + x1α + x2α2 + x3α3 had opposite in ring Z[α], that is that ξ–1 ∈ Z[α] – that ξ would be a divider of unit. Method of work: we design (basing on linear operators) subring M (m3, m2, m1, m0) of rings of matrixes M[4x4](Z). It subring is isomorphying to ring Z[α]. The fact is used: „ the matrix with the integer elements A ∈ Z[mxm](Z) has an opposite matrix in only case when its determinant |A | = ± 1. From property of isomorphism: “if φ(ξ) = Aξ isomorphism of ring Z[α] in ring M (m3, m2, m1, m0), so (φ(ξ))–1 = φ(ξ–1) = Aξ–1 = (Aξ )–1 “ follows, that ξ ∈ Z[α] has opposite ξ–1 ∈ Z[α] in only case when a matrix Aξ has an opposite matrix. And a matrix Aξ has an opposite matrix in a ring M (m3, m2, m1, m0) in only case when its determinant |A| = ± 1.
| Identifer | oai:union.ndltd.org:LABT_ETD/oai:elaba.lt:LT-eLABa-0001:E.02~2004~D_20040610_174317-66316 |
| Date | 10 June 2004 |
| Creators | Falkevič, Irena |
| Contributors | Dzemyda, Gintautas, Lipeikienė, Joana, Leonavičius, Gražvydas, Januškevičius, Romanas, Survila, Pranas, Šinkūnas, Juozas, Vilnius Pedagogical University |
| Publisher | Lithuanian Academic Libraries Network (LABT), Vilnius Pedagogical University |
| Source Sets | Lithuanian ETD submission system |
| Language | Lithuanian |
| Detected Language | English |
| Type | Master thesis |
| Format | application/pdf |
| Source | http://vddb.library.lt/obj/LT-eLABa-0001:E.02~2004~D_20040610_174317-66316 |
| Rights | Unrestricted |
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