研FMIS-Ch1-AlgebraicPreliminaries [兼容模式]

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Definition of Group
Definition 1.1.2 A group (群), denoted by (G, ), or simply G, is a nonempty set G of elements together with a binary operation , such that the following three properties hold: (1) Associativity: For any a, b, cG, a(bc)= (ab)c. (2) Existence of identity (恒等元)(or unity: 单位元): There is a unique element eG, called the identity, such that ea= ae=a, aG. (3) Existence of inverse (逆元): For every aG there is a unique element bG such that ab= ba=e. This b is denoted by a1 and called the inverse of a.
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Definition of Group
The operation rules For any n, mZ, aG,
Multiplicative Notation an=(a1)n anam =an+m (an)m =anm a0=1 Additive Notation (n)a=n(a) na+ma=(m+n)a m(na)=(mn)a 0a=0
西南交通大学 2013级硕/博研究生课程
有限域与计算数论
Fundamentals of Mathematics in Information Security
信息科学与技术学院
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Part 1 Algebraic Foundations
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Chapter 1 Algebraic Preliminaries 1.1 Groups 1.2 Rings 1.3 Fields
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Definition of Ring
Definition 1.2.5 An division ring(除环) is a ring R with identity 1 0 that satisfies: for each a 0R, the equation ax=1 and xa=1 have solutions in R. Example 1.2.1 (Z,+,), (Q,+,), (R,+,) and (C,+,) are all rings The even integers (Ze,+,) form a commutative ring without identity. Let (R, +) be any Abelian group. Define ab =0 for all a, bR, then (R, +, ) is a ring.
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Definition of Group
Definition 1.1.2 If the group (G, ) also satisfies (4) Commutativity: For any a, bG, ab=ba, then the group is called a commutative group (交换群) (or Abelian group).
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Definition of Group
Definition 1.1.1 A binary operation (二元运算) on a set S is a rule that assigns to each ordered pair (a, b) of elements of S a unique element of S.
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Chapter 1 Algebraic Preliminaries
1.1 Groups 1.2 Rings 1.3 Fields
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Definition of Ring
Definition 1.2.1 A ring (环), denoted by (R, +, ), or simply R, is a set R, together with two binary operation + and , which we call addition and multiplication, such that the following properties hold: (1) R is a commutative group with respect to +; (2) Associative law holds for : a(bc)= (ab)c, a, b, cR. (3) The distributive law holds : a(b+c)= ab+ac, a, b, cR, (b+c)a = ba+ca, a, b, cR.
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Definition of Group
Example 1.1.2 (Z,+), (Q,+), (R,+) and (C,+) are commutative groups (Z+, +) is not a group: since there is no identity element for + in Z+. (Z+, ) is not a group: there is an identity element 1, but no inverse of 3. (Q+, ) and (R+, ) are commutative groups (Q, ), (R, ) and (C, ) are not groups (Q*, ), (R*, ) and (C*, ) are commutative groups
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Finite Group, Order, Subgroup
Definition 1.1.3 A group is called a finite group (有限 群) if it has a finite number of elements; otherwise it is called a infinite group (无限群). Definition 1.1.4 The order (阶) of a group G, denoted by | G | (or by #(G)), is the number of elements in G. Example The order of Z is |Z|=.
Example 1.1.1 Ordinary addition + is a binary operation on the set N, Z, R, C. Ordinary multiplication is another binary operation on the same sets.
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Finite Group, Order, Subgroup
Definition 1.1.5 A nonempty set H of a group G which is itself a group, under the same operation, is called a subgroup (子群) of G. Subgroups of G other than trivial subgroup {e} and G itself are called nontrivial subgroup of G. Example Z is a subgroup of Q.
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The order of the element
Definition 1.1.7 Let a be an element of a group G. If <a> is finite, then its order is called the order of the element a. Otherwise, a is called an element of infinite order.
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Definition of Ring
Definition 1.2.2 A commutative ring(交换环) is a ring that further satisfies: ab = ba, a, bR. Definition 1.2.3 A ring with identity is a ring that contains an element 1 satisfying: a1 = 1a = a, a R. Definition 1.2.4 An integral ring (整环) is a commutative ring with identity10 that satisfies: a, bR ab=0 a =0 or b=0. Remark: There are no zero divisors.
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Notations
Notations The set of natural numbers: N={1, 2, 3,…} The set of integers: Z={0, 1, 2, 3,…}; Z+= {1, 2, 3,…}= N; Z*={aZ: a 0}={1, 2, 3,…} ; The set of all residue classes modulo a positive integer n: Zn=Z/nZ ={0,1,2,…, n1}. The set of rational numbers: Q={a/b: a,bZ and b0}; Q+={aQ: a>0}; Q*={aQ: a 0}.