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Group (G)
A group G, denoted by { G, + } such that (a, b) of elements in G an element (a + b) in G × G
Following axioms are obeyed:
- Closure under addition: If a and b belong to G, then a + b is also in G
- Associative of addition: a + (b + c) = (a + b) + c for all a, b, c in G
- Identity element: There is an element e in G such that a + e = e + a = a for all a in G
- Inverse element: For each a in G, there is an element a′ in G such that a + a′ = a′ + a = e
Abelian Group
The group in which the commutative law is established is called the Abelian group
- Commutative of addition: a + b = b + a for all a, b in G
Cyclic Group (Always Abelian)
The group is cyclic if every element of G is a power ak (k is an integer) of a fixed element a ∈ G
Ring (R)
A ring R, denoted by { R, +, × }, such that for all a, b, c in R
Simply, a ring is a set of elements in which we can do addition, subtraction (Ex. a - b = a + (-b)),
and multiplication
Following axioms are obeyed:
- Closure under multiplication: If a and b belong to R, then ab is also in R
- Associativity of multiplication: a(bc) = (ab)c for all a, b, c in R
- Distributive laws: a(b + c) = ab + ac, (a + b)c = ac + bc for all a, b, c in R
Commutative Ring
A Commutative Ring works in an integral domain
Following axioms are obeyed:
- Commutativity of multiplication: ab = ba for all a, b in R
- Multiplicative identity: There is an element 1 in R such that a×1 = 1×a = a for all a in R
- No zero divisors: If a, b in R and ab = 0, then either a = 0 or b = 0
Field (F)
A field F, denoted by { F, + , × }, such that for all a, b, c in F
Simply, a field is a set of elements in which we can do addition, subtraction, multiplication, and division without
leaving the set
Division is defined with the following rule: a/b = a(b-1)
Following axioms are obeyed:
- Multiplicative inverse: For each a in F, except 0, there is an element a-1 in F such that aa-1 = (a-1)a = 1
Finding Multiplicative Inverse
If n is prime, then all of the nonzero integers in Zn are relatively prime to n
Ex)
n=11 (since 11 is a prime number): Z11 = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
GCD of 1 and 11: GCD is 1
GCD of 2 and 11: GCD is 1
GCD of 3 and 11: GCD is 1
GCD of 4 and 11: GCD is 1
GCD of 5 and 11: GCD is 1
GCD of 6 and 11: GCD is 1
GCD of 7 and 11: GCD is 1
GCD of 8 and 11: GCD is 1
GCD of 9 and 11: GCD is 1
GCD of 10 and 11: GCD is 1
➤ All of these nonzero integers in Z11 have a GCD of 1 with 11, which means they are all relatively prime to 11
This illustrates the general principle that if n is a prime number, then every nonzero element of Zn will be relatively prime to n
To find the multiplicative inverses of all nonzero integers in Zn, we need to find an integer x for each nonzero integer a in Zn such that a × x ≡ 1 (mod n)
The multiplicative inverse of 1 is 1 (since 1 × 1 ≡ 1 mod 11)
The multiplicative inverse of 2 is 6 (since 2 × 6 ≡ 1 mod 11)
The multiplicative inverse of 3 is 4 (since 3 × 4 ≡ 1 mod 11)
The multiplicative inverse of 4 is 3 (since 4 × 3 ≡ 1 mod 11)
The multiplicative inverse of 5 is 9 (since 5 × 9 ≡ 1 mod 11)
The multiplicative inverse of 6 is 2 (since 6 × 2 ≡ 1 mod 11)
The multiplicative inverse of 7 is 8 (since 7 × 8 ≡ 1 mod 11)
The multiplicative inverse of 8 is 7 (since 8 × 7 ≡ 1 mod 11)
The multiplicative inverse of 9 is 5 (since 9 × 5 ≡ 1 mod 11)
The multiplicative inverse of 10 is 10 (since 10 × 10 ≡ 1 mod 11)
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