36 terms

Ring

A set R of complex numbers is a ring if

1. x+y,x-y,xy are in R for all x,y in R.

2. 0,1 in R

1. x+y,x-y,xy are in R for all x,y in R.

2. 0,1 in R

Factor in a ring

Let a,b be elements of a ring R. We say a is a factor of b in R if there exists a z in R such that b=az.

Unit

An element u of a ring R is a unit if u is a factor of 1 in R. i.e. if 1/u is in R.

Units of Z[i]

1,-1,i,-i

Associate

Let x be an element of a ring R. The associates of x are the elements of R of the form ux where u is a unit in R.

Irreducible

A non-zero, non-unit element x of R is irreducible if the only factors of x in R are the units of R and the associates of x in R.

The norm of a

The norm of a N(a)=aa** where a** is the complex conjugate of a.

N:Z[i]->Z

N:Z[i]->Z

Euclidean Ring

A ring R of complex numbers is a Euclidean Ring (ER) if

1. |a|^2 is in R for every a in R.

2. Given a fraction of the form

z=a/d where a,d in R and d not equal to 0, there exists a q in R such that |z-q|^2<1.

1. |a|^2 is in R for every a in R.

2. Given a fraction of the form

z=a/d where a,d in R and d not equal to 0, there exists a q in R such that |z-q|^2<1.

Highest common factor

Let a,b be in R. A highest common factor (HCF) of a and b is a H in R such that

1. H|a and H|b

2. If there exists a x such that x|a and x|b then x|H

1. H|a and H|b

2. If there exists a x such that x|a and x|b then x|H

Prime

A non zero non unit element p in a ring R is prime if for all a,b in R, p|ab implies that p|a or p|b

Unique Factorization Domain

A ring R is a Unique Factorization Domain (UFD) if for every non zero non unit a in R,

1. There exist irreducibles p1,p2,....,pn in R such that a=p1p2...pn, and

2. If there are irreducibles q1,q2,...,qn in R such that a=q1q2...ql then l=n and there exists a one-to-one correspondence between the ps and the qs so that corresponding numbers are associate.

1. There exist irreducibles p1,p2,....,pn in R such that a=p1p2...pn, and

2. If there are irreducibles q1,q2,...,qn in R such that a=q1q2...ql then l=n and there exists a one-to-one correspondence between the ps and the qs so that corresponding numbers are associate.

Order of a modulo p

Let p be a prime and a an integer with a not congruent to 0 modulo p. The order of a modulo p is the smallest natural number d such that

a^d is congruent to 1 modulo p.

a^d is congruent to 1 modulo p.

Primitive Root

Let p be a prime. A primitive root modulo p is an integer z with order p-1 modulo p.

f(X) is congruent to g(X) modulo n

Let f(X) and g(X) be polynomials with integer coefficients and let n be a natural number. Then we say that f(X) is congruent to g(X) modulo n if the coefficients of each power of X are congruent modulo n.

Degree modulo n of f(X)

Let n be a natural number and f(X) be a polynomial with integer coefficients. Then the degree modulo n of f(X) is the largest d such that the coefficient of X^d is not congruent to 0 modulo n.

Lagrange's Theorem

Let p be a prime and f(X) be a polynomial with integer coefficients and d the degree modulo p with d >= 0. Then the equation

f(X) congruent to 0 modulo p

has at most d distinct solutions modulo p.

f(X) congruent to 0 modulo p

has at most d distinct solutions modulo p.

Quadratic residue modulo p

Let a be an integer and p a prime where a is NOT congruent to 0 modulo p. Then a is a quadratic residue modulo p if there exists an integer x such that

x^2 is congruent to a modulo p.

If there doesn't exist such an x then a is a quadratic non-residue.

x^2 is congruent to a modulo p.

If there doesn't exist such an x then a is a quadratic non-residue.

Legendre symbol (a/p)

(a/p)=

1 if a is a quadratic residue

-1 if a is a quadratic nonresidue

0 if a is congruent to 0 modulo p

1 if a is a quadratic residue

-1 if a is a quadratic nonresidue

0 if a is congruent to 0 modulo p

Euler's Criterion

Let a be an integer and p be an ODD prime. Then

(a/p) is congruent to a^[1/2(p-1)] modulo p.

(a/p) is congruent to a^[1/2(p-1)] modulo p.

Least Residue modulo p of a

Let a be an integer and let p be an odd prime. The least residue modulo p of a is the unique integer b such that

1. -1/2(p-1) =< b =< 1/2(p-1) and

2. b is congruent to a modulo p

1. -1/2(p-1) =< b =< 1/2(p-1) and

2. b is congruent to a modulo p

Gauss' Lemma

Let p be an odd prime and a be an integer with a not congruent to 0 mod p. Let u be the number of numbers

1a, 2a, ..., 1/2(p-1)a

that have a negative least residue. Then

(a/p)=(-1)^u

1a, 2a, ..., 1/2(p-1)a

that have a negative least residue. Then

(a/p)=(-1)^u

(2/p)

(2/p)=

1 if p is congruent to +-1 modulo 8

-1 if p is congruent to +-3 modulo 8

1 if p is congruent to +-1 modulo 8

-1 if p is congruent to +-3 modulo 8

Gauss' Law of Quadratic Reciprocity

Let p,q be distinct odd primes and integers. Then

(p/q)=(-1)^[1/2(p-1) * 1/2(q-1)].

i.e. (p/q) = (q/p) unless p is congruent to q is congruent to 3 modulo 4, in which case

(p/q)=-(q/p).

(p/q)=(-1)^[1/2(p-1) * 1/2(q-1)].

i.e. (p/q) = (q/p) unless p is congruent to q is congruent to 3 modulo 4, in which case

(p/q)=-(q/p).

List of irreducible Gaussian integers

1. 1+i

2. prime numbers p with p congruent to 3 modulo 4

3. the irred. factors of the prime numbers p where p congruent to 1 modulo 4.

4. all the associates of the above

2. prime numbers p with p congruent to 3 modulo 4

3. the irred. factors of the prime numbers p where p congruent to 1 modulo 4.

4. all the associates of the above

Wallis' Equation

x^4 + 9 = y^2

Coprime in a UFD

Two elements of a UFD R are coprime if their only common factors are units.

Theorem: The only integer solutions to

y^3=x^2+2 are (x,y)=(+-5,3)

y^3=x^2+2 are (x,y)=(+-5,3)

Steps of Proof:

Step 1. Prove y odd (consider congruences modulo 4)

Step 2. x+sqrt{-2} and x-sqrt{-2} coprime (let a be a common factor- factor of difference)

Step 3. Show x+sqrt{-2} a cube power

(x+sqrt{-2} = un^3)

Step 4. (x,y)=(+-5,3)

Step 1. Prove y odd (consider congruences modulo 4)

Step 2. x+sqrt{-2} and x-sqrt{-2} coprime (let a be a common factor- factor of difference)

Step 3. Show x+sqrt{-2} a cube power

(x+sqrt{-2} = un^3)

Step 4. (x,y)=(+-5,3)

ord_p(a)

Let a be a non zero element of R, a UFD, and p be an irreducible element in R. Then ord_p(a) is defined as the number of associates of p that appear in the factorization of a into irreducibles.

ord_p(a) where a is a unit

0

Modular arithmetic in a ring definition

Let a,b be in R, a ring, and n be a non zero element of R. Then we say a is congruent to b modulo n if n is a factor of (a-b) in R.

Complete set of residues in a ring

A complete set of residues modulo n is a subset S of a ring R such that every element of R is congruent modulo n to exactly one element of S.

Fermat's little theorem in Gaussian integers

Let p be a prime number and a be in Z[i]. Then

a^p congruent to

->a mod p if p congruent to +-1 mod 4

->a* mod p if p congruent to +-3 mod 4

a^p congruent to

->a mod p if p congruent to +-1 mod 4

->a* mod p if p congruent to +-3 mod 4

Fermat's little theorem in R=Z[sqrt3]

Let p be a prime number and a be in Z[sqrt3].

Then a^p congruent to

->a mod p if p congruent to +-1 mod 12

->a* mod p if p congruent to +- 5 mod 12

Then a^p congruent to

->a mod p if p congruent to +-1 mod 12

->a* mod p if p congruent to +- 5 mod 12

nth Mersenne number

Mn= 2^n -1

Theorem (Lucas Test)

Define a sequence ri by r0=4 and ri+1=(ri)^2-2.

Let n>=3 be a prime.

Mn is prime iff r(n-2) is congruent to 0 mod Mn.

aka Mn is a factor of r(n-2)

Let n>=3 be a prime.

Mn is prime iff r(n-2) is congruent to 0 mod Mn.

aka Mn is a factor of r(n-2)

Order in a ring

Let R be a ring, a be a non zero element of R and n be in R. Then the order of a modulo n in R is the smallest natural number d such that

a^d is congruent to 1 modulo n.

a^d is congruent to 1 modulo n.