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Quaternion algebras
Bečka, Pavel ; Klaška, Jiří (referee) ; Kureš, Miroslav (advisor)
This thesis deals with quaternion algebras. A quaternion algebra is a four dimensional vector space with basis 1, i, j, k and multiplication defined as i2 = a, j2 = b, ij = -ji = k. The thesis deals with the basic attributes of quaternion algebras, quaternion orders and maximal orders. Lastly the thesis deals with the concept of discriminant of algebras and connected terms like Hilbert symbol and Legendre symbol. Throughout the thesis we show solved problems using mathematical software SAGE.
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Maximal Orders
Tlustá, Stanislava ; Příhoda, Pavel (advisor) ; Růžička, Pavel (referee)
Maximal Orders Stanislava Tlustá Abstract This thesis summarizes basic properties of lattices and orders over Dedekind domain in separable algebras. The concepts of reduced norm and reduced trace are introduced and applied to few examples of rational algebras. By that the maximal orders are found. The properties of maximal orders are stated and used to explore new types of ideals: normal ideals and Λ-ideals. At the end of this thesis the isomorphisms of lattices are examined and the Jordan-Zassenhaus theorem is proved. 1
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Quaternion algebras
Bečka, Pavel ; Klaška, Jiří (referee) ; Kureš, Miroslav (advisor)
This thesis deals with quaternion algebras. A quaternion algebra is a four dimensional vector space with basis 1, i, j, k and multiplication defined as i2 = a, j2 = b, ij = -ji = k. The thesis deals with the basic attributes of quaternion algebras, quaternion orders and maximal orders. Lastly the thesis deals with the concept of discriminant of algebras and connected terms like Hilbert symbol and Legendre symbol. Throughout the thesis we show solved problems using mathematical software SAGE.
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