Theorems with a name

Question 1

Eulers Lemma

Answer 1

If a | bc and gcd(a,b)=1 then a | c .

Question 2

Fundamental Theorem of Arithmetic

Answer 2

Every positive integer n is a product of prime numbers and it’s factorisation into primes is unique up to the order of factors.

Question 3

Euclids Theorem

Answer 3

The number of primes is infinite.

Question 4

Euler’s Theorem

Answer 4

m positive in Z.

a an integer coprime to m.

Then a^φ(m) ≡ 1 (mod m) .

Question 5

Fermats Little Theorem

Answer 5

Let p be prime.

1. Let a an integer not divisible by p.
   Then a^p-1 ≡ 1 (mod p) .
2. For all integers a,
   a^p ≡ a (mod p) .

Question 6

Wilson’s Theorem

Answer 6

Let p be a prime.

Then (p-1)! ≡ -1 (mod p)

Question 7

Chinese Remainder Theorem 1

Answer 7

Let m, n ϵ |N be coprime.

Then for every pair of integers a, b the simultaneous congruences

• x ≡ a (mod m)*
• x ≡ b (mod n)*

have a solution which is unique modulo mn.

More generally, if d = gcd(m,n) then the congruences have a solution ⇔ a ≡ b (mod d) and the solution is unique modulo lcm(m,n) = mn/d.

Question 8

Chinese Remainder Theorem 2

Answer 8

Let m, n be coprime.

Then Z/mnZ ≅ Z/mZ X Z/nZ .

In units, Umn ≅ U_m X U_n .

Question 9

Euler’s Criterion

Answer 9

(a/p) ≡ a^(p-1)/2 (mod p)

Question 10

Gauss’s Lemma

Answer 10

Let p be an odd prime and a an integer not divisible by p.

Then

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

where s is the number of integers i with 0 < i < p/2 for which the least residue of ai is negative.

Question 11

Quadratic Reciprocity

Answer 11

Let p and q be distinct odd primes.

Then (p/q)(q/p) = (-1)^(p-1)(q-1)/2

so (q/p) = (p/q) , if p ≡ 1 or q ≡ 1 (mod 4)

and (q/p) = - (p/q) , if p ≡ q ≡ 3 (mod 4) .

Question 12

Minkowski’s Theorem

Answer 12

Let L ≤ Zⁿ be a lattice of index m.
Let S⊆|Rⁿ be a bounded, convex, symmetric domain.

If S has volume v(S) > 2ⁿ m , then S contains a non-zero element of L.

Same conclusion holds when v(S) = 2ⁿ m provided S is compact.

Question 13

Lagrange’s Theorem

Answer 13

Every positive number can be expressed as a sum of four squares.

Question 14

Fermats Last Theorem

Answer 14

Let n ≥ 3.

Then there are no solutions in positive integers to:

xⁿ + yⁿ = zⁿ .

Question 15

Ostrowski’s Theorem

Answer 15

Every non-trivial norm on (Q is equivalent either to the standard absolute value |x| or to the p-adic norm |x|_p for some prime p.

All these norms are inequivalent.

Question 16

Hensel Lifting Theorem

Answer 16

Let F(x) ϵ Z_p[x] be a polynomial.

Let x₁ ϵ Z_p satisfy f(x₁) ≡ (mod p) and f’(x₁) ≡ 0 (mod p).

Then there exists a unique x ϵ Z_p such that:

f(x) = 0

and x ≡ x₁ (mod p)