Number Theory: An Introduction to Mathematics

66 I The Expanding Universe of Numbers for someα 1 ,...,αm ∈D.Thecoefficientsα 1 ,...,αmneed not be uniquely deter- mined. Evide ...

9 Vector Spaces and Associative Algebras 67 IfVhas a basis containingnelements, we sayVhasdimension nand we write dimV=n. We say ...

68 I The Expanding Universe of Numbers I :V →Vdefined byIv =vfor everyv ∈Vis clearly linear. If a linear map T:V→Wis bijective, ...

9 Vector Spaces and Associative Algebras 69 the ring structure, we can regardAas a vector space overF. The associative algebra i ...

70 I The Expanding Universe of Numbers j=γ+δb,whereγ,δ∈Randδ=0. Sincej^2 =γ^2 + 2 γδb+δ^2 α=−1, we must haveγ=0. Thusj=δbandji= ...

10 Inner Product Spaces 71 which impliesaib=bai(i= 1 ,...,n). Since this holds for allb∈MandMis a maximal subfield ofD, it follo ...

72 I The Expanding Universe of Numbers Thus‖v‖≥0, with equality if and only ifv=O. Evidently ‖αv‖=|α|‖v‖ for allα∈Fandv∈V. Inner ...

10 Inner Product Spaces 73 The norm in any inner product spaceVsatisfies theparallelogram law: ‖u+v‖^2 +‖u−v‖^2 = 2 ‖u‖^2 + 2 ‖v ...

74 I The Expanding Universe of Numbers Since‖w‖^2 =|γ 1 |^2 +···+|γm|^2 , this yieldsBessel’s inequality: |〈v,e 1 〉|^2 +···+|〈v, ...

11 Further Remarks 75 Analmost periodic function, in the sense of Bohr (1925), is a functionf:R→C which can be uniformly approxi ...

76 I The Expanding Universe of Numbers Boolean algebra was given by Huntingdon [39]. For an introduction to Stone’s repre- senta ...

11 Further Remarks 77 the roots of a quartic discovered by his pupil Ferrari, was the most significant Western contribution to m ...

78 I The Expanding Universe of Numbers Of wider significance are the associative algebras introduced in 1878 by Clifford [15] (p ...

12 Selected References 79 Field theory was established as an independent subject of study in 1910 by Steinitz [68]. The books of ...

80 I The Expanding Universe of Numbers [9] H. Bohr,Almost periodic functions, English transl. by H. Cohn and F. Steinhardt, Chel ...

12 Selected References 81 [38] J.E. Humphreys,Reflection groups and Coxeter groups, Cambridge University Press, 1990. [39] E.V. ...

82 I The Expanding Universe of Numbers [71] A. Sudbery, Quaternionic analysis,Math. Proc. Cambridge Philos. Soc. 85 (1979), 199– ...

II Divisibility.................................................. 1 GreatestCommonDivisors In the setNof all positive integers w ...

84 II Divisibility Proposition 1Any a,b∈Nhave a greatest common divisor(a,b). Proof Without loss of generality we may supposea ≥ ...

1 Greatest Common Divisors 85 If we put A=([a,b],[a,c]), B=[a,(b,c)], then by what we have already proved, A=(ab/(a,b),ac/(a,c)) ...