Title: Theory / Unimportant / Math / Number Theory / Ireland, Rosen/
Tags: fromMath mathToDef mathToSee automaticImplies TypToQuestion defToAlgebra examplesSwitchExample1 setAdditionMultiplication toBinaryRelation typToBinary

Definition: The set of relative integers
Page: 001
Chapter: 1
Notation: \(\mathbb Z\)

Name: \(a\) divides \(b\)
Definition: There exists an integer \(c\) such that \(ac=b\).
Page: 001
Chapter: 1
Variables: \(a,b\in\mathbb Z\)
Notation: \(a|b\)

Q: [...] implies there is a [...] factorization
[...].
A: \(n\ne0\); prime; \(n=(-1)^{\epsilon(n)}\Pi_ip^{a(p)}\), with the exponents uniquely determined by \(n\)
Page: 003
Chapter: 1
Variables: \(n\in\mathbb Z\).
Kind: Theo

Smaller0: \(\epsilon(z)\)
Greater0: 1 if \(z<0\) and \(0\) otherwise
Page: 003
Chapter: 1
Variables: \(z\in\mathbb Z\)
Smaller to greater: equals
Hide middle: yes

Definition: \(q,r\in\mathbb Z\) such that \(a=qb+r\) with \(0\le r<b\).
Page: 004
Chapter: 1
Variables: \(a,b\in\mathbb Z\) and \(b>0\)
Index: 2
Kind: Lemma

Definition: The set of integers of the form \(\sum a_ix_i\) with \(x_i\in\mathbb Z\).
Page: 004
Chapter: 1
Variables: \(a_1,\dots,a_n\in\mathbb Z\).
Notation: \((a_1,\dots,a_n)\)
Kind: Def

Name: Two relatively prime elements of \(\mathbb Z\).
Definition: Two integers whose only common divisors are \(\pm 1\), the units.
Page: 005
Chapter: 1
Kind: Def

Smaller0: \(a|bc\)
Greater0: \(a|c\)
Page: 005
Chapter: 1
Variables: \(a,b,c\in\mathbb Z\).
Kind: Proposition
Smaller1: \((a,b)=1\)
Connect Smaller0: and that

Smaller0: \(a|bc\)
Greater0: \(a|c\) or \(a|b\)
Page: 005
Chapter: 1
Variables: \(a,b,c\in\mathbb Z\).
Kind: Cor
Hide middle: yes
Smaller1: \(a\) is prime
Connect Smaller0: and

Name: The order of \(n\) at \(p\)
Definition: The greatest integer \(i\) such that \(p^i|n\).
Page: 003
Chapter: 1
Variables: \(n,p\in\mathbb Z\) with \(p\ne0\).
Notation: \(ord_pn\)

Smaller0: \(\text{ord}_pab\)
Greater0: \(\text{ord}_pa+\text{ord}_pb\)
Page: 005
Chapter: 1
Variables: \(a,b\in\mathbb Z\)
\(p\) prime
Index: 2
Kind: Cor
Smaller to greater: =

Definition: The set of polynomials with coefficient in \(k\).
Page: 006
Chapter: 1
Section: 2
Variables: \(k\) be a field.
Notation: \(k[x]\)

Name: \(f\) divides \(g\)
Definition: There exists a polynomial \(h\) such that \(fh=g\).
Page: 006
Chapter: 1
Section: 2
Variables: \(k\) be a fieald
\(f,g\in k[x]\).

Smaller0: \(f|g\) and \(g|f\)
Greater0: there exists a non zero \(c\in k\) such that \(f=cg\)
Page: 006
Chapter: 1
Section: 2
Variables: \(k\) be a fieald. Let \(f,g\in k[x]\).
Smaller to greater: iff
Hide middle: yes

Smaller0: \(f\) divides all polynomial
Greater0: \(f\) is a constant
Page: 006
Chapter: 1
Section: 2
Variables: \(k\) be a field
\(f\in k[x]\).
Smaller to greater: iff

Name: An irreducible polynomial
Definition: \(f\in k[x]\) such that, for all \(q\) which divides \(f\), either \(q\) is a constant, or it is a constant multiple of \(f\).
Page: 006
Chapter: 1
Section: 2
Variables: \(k\) be a fieald.

Q: There is [...] way to write \(p\) as [...].
A: exactly one; the product of irreducible monic polynomials and of a constant
Page: 006
Chapter: 1
Section: 2
Variables: \(p\in k[x]\)
Index: 2, 1
Kind: Theo, Lem

Name: A monic polynomial
Definition: A polynomial whose leading coefficient is 1.
Page: 006
Chapter: 1
Section: 2

Name: The order of \(p\) at \(f\)
Definition: The greatest integer \(i\) such that \(p^i|f\).
Page: 007
Chapter: 1
Section: 2
Variables: \(p\) be a monic irreduccible polynomial and \(f\) be a polynomial.
Notation: \(ord_p(f)\)

Q: There exists:
-[...].
-[...].
A: polynomials \(h,r\in k[x]\) such that \(f=gh+r\), where the degree of \(r\) is less than the degree of \(g\); \(d\) a greatest commun divisor of \(a\) and \(b\). With \((f,g)=(d)\)
Page: 007
Chapter: 1
Section: 2
Variables: \(f,g\in k[x]\).
Kind: Lemma

Q: [...].
A: The set of polynomials of the form \(\sum f_ih_i\) with \(h_i\in k[x]\)
Page: 007
Chapter: 1
Section: 2
Variables: \(f_1,\dots,f_n\in\mathbb K[x]\).
Kind: Def
Notation: \((f_1,\dots,f_n)\)

Name: Two polynomials which are relatively prime
Definition: \(f,g\in k[x]\) such that \((f,g)=(1)\).
Page: 008
Chapter: 1
Section: 2
Variables: \(k\) a field.

Smaller0: relatively prime
Greater0: \(f|h\)
Page: 008
Chapter: 1
Section: 2
Variables: \(f,g,h\in k[x]\).
Kind: Proposition
Smaller1: \(f|gh\)
Prefix Smaller: \(f,g\) are
Connect Smaller0: and

Smaller0: irreducible
Greater0: \(p|gh\)
Page: 008
Chapter: 1
Section: 2
Variables: \(p\in k[x]\).
Kind: Cor
Hide middle: yes
Greater1: \(p|g\) or \(p|h\)
Connect Greater0: then
Prefix Smaller: \(p\) is

Smaller0: \(ord_f(gh)\)
Greater0: \(ord_f(g)+ord_f(h)\)
Page: 008
Chapter: 1
Section: 2
Variables: \(f\in k[x]\).
Kind: Cor
Smaller to greater: =
Hide middle: yes
Assuming: \(f\) is irreduccible

Q: [...].
A: An integer domain \(R\) such that there is a function \(\lambda\) from nonzero elements of \(R\) to the set non-negative integers such that if \(a,b\in R\), \(b\ne0\) there exists \(c,d\in R\) with the property \(a=cb+d\) and either \(d=0\) or \(\lambda(d)<\lambda(b)\)
Page: 008, 009
Chapter: 1
Section: 3
Kind: Definition
Name: Euclidean domain
Q: [...] and [...].
A: \(\mathbb Z\); \(k[x]\)

Name: A principal ideal domain
Definition: An ideal such that each ideal is principal.
Page: 009
Chapter: 1
Section: 3
Variables: \(R\) an Euclidean domain.
Index: 2
Kind: Definition
Abbreviation: PID
French: Anneau principal

Name: A unit
Definition: A divisor of \(1\).
Page: 002
Chapter: 1
Section: 3
Variables: \((R,1,\times)\) a monoid.
Kind: Definition

Name: Two associate elements of \(R\)
Definition: \(a\) and \(au\), for some \(a\in R\) and some unit \(u\).
Page: 009
Chapter: 1
Section: 3
Variables: \((R,1,\times)\) a monoid and \(a,b\in R\).
Index: 2
Kind: Definition
Definition2: Two elements \(a\) and \(b\) such that \((a)=(b)\)

Name: An irreduccible element of \(R\)
Definition: An element \(a\) such that, for all \(b\), \(b|a\) implies that \(b\) is either a unit or associate to \(a\).
Page: 009
Chapter: 1
Section: 3
Variables: \((R,1,\times)\) a monoid and \(a\in R\).
Kind: Definition

Smaller0: \(a|b\)
Greater0: \((a)\subseteq (b)\)
Page: 009
Chapter: 1
Section: 3
Variables: \((R,1,\times)\) a monoid and \(a,b\in R\).
Index: 2
Kind: Definition
Smaller to greater: iff

Smaller0: \(a\) is a unit
Greater0: \((a)=R\)
Page: 009
Chapter: 1
Section: 3
Variables: \((R,1,\times)\) a monoid and \(a\in R\).
Index: 2
Kind: Definition
Smaller to greater: iff
Hide middle: yes

Smaller0: \(a\) and \(b\) are associate
Greater0: \((a)=(b)\)
Page: 009
Chapter: 1
Section: 3
Variables: \((R,1,\times)\) a monoid
\(a,b\in R\).
Index: 2
Kind: Definition
Smaller to greater: iff
Hide middle: yes

Smaller0: \(a\) is prime
Greater0: \(bc\in (a)\) implies that either \(b\in(a)\) or \(c\in (a)\)
Page: 009
Chapter: 1
Section: 3
Variables: \((R,1,\times)\) a monoid and \(a\in R\).
Index: 2
Kind: Definition
Smaller to greater: iff
Hide middle: yes

Name: The set of greatest common divisors of \(a\) and \(b\)
Definition: The elements \(d\) such that \(d|a\), \(d|b\) and for all \(d'\), if \(d'|a\) and \(d'|b\) then \(d'|d\).
Page: 009
Chapter: 1
Section: 3
Variables: \((R,1,\times)\) a monoid
\(a,b\in R\).
Index: 2
Kind: Definition

Smaller0: \(R\) is a PID
Greater0: a gcd \(d\) and \((a,b)=(d)\)
Page: 010
Chapter: 1
Section: 3
Variables: \((R,1,\times)\) a monoid
\(a,b\in R\).
Index: 2
Kind: Definition
Hide middle: yes
Prefix Greater: \(a\) and \(b\) have
Extra: Prop 2

Name: Relatively prime elements of \(R\)
Definition: Two elements whose only common divisors are units.
Page: 010
Chapter: 1
Section: 3
Variables: \((R,1,\times)\) a monoid

Smaller0: \(R\) is a PID and \(a,b\) are relatively prime
Greater0: \((a,b)=R\)
Page: 010
Chapter: 1
Section: 3
Variables: \((R,1,\times)\) a monoid and \(a,b\in R\).
Kind: Cor

Smaller0: \(a\) is irreduccible
Greater0: \(a\) is prime
Page: 010
Chapter: 1
Section: 3
Variables: \((R,1,\times)\) a monoid and \(a\in R\).
Index: 2
Kind: Cor
Hide middle: yes
Assuming: \(R\) is a PID

Smaller0: \(R\) is a PID and \((a_i)\subseteq(a_{i+1})\)
Greater0: the sequence is ultimately constant
Page: 010
Chapter: 1
Section: 3
Variables: \((R,1,\times)\) a monoid
\((a_i)_{i\in\mathbb N}\in R\).
Kind: Lemma
Hide middle: yes

Smaller0: \(R\) is a PID and \(a\ne0\) is nonunit
Greater0: a product of irreducibles
Page: 011
Chapter: 1
Section: 3
Variables: \((R,1,\times)\) a monoid
\(a\in R\)
Kind: Proposition
Hide middle: yes
Prefix Greater: \(a\) is equal to

Definition: The greatest integer \(i\) such that \(a^i|b\) and \(a^{i+1}\nmid b\).
Page: 011
Chapter: 1
Section: 3
Variables: \((R,1,\times)\) a monoid
\(a,b\in R\)
Notation: \(ord_a(b)\)
Index: 2
Kind: Lemma
Assuming: \(R\) is A PID

Smaller0: \(ord_p(ab)\)
Greater0: \(ord_p(a)+ord_p(b)\)
Page: 011
Chapter: 1
Section: 3
Variables: \((R,1,\times)\) a PID
\(a,b,p\in R\) with \(p\) prime
Kind: Lemma
Smaller to greater: =

Smaller0: \(S\) is a set of primes, all primes of \(R\) have an associate in \(S\), and no two primes of \(S\) are associate
Greater0: \(a=u\Pi_{p}p^{e(p)}\), where the choice of \(u\) unit, \(p\in S\) and \(e(p)\in\mathbb N\) is unique. With \(e(p)=ord_p(a)\)
Page: 012
Chapter: 1
Section: 3
Variables: \(R\) a PID, \(a\in R\) nonzero, and \(S\subseteq R\).
Kind: Theorem
Hide middle: yes

Q: [...] is defined as [...].
A: \(\mathbb Z[i]\); the set of elements of the form \(a+ib\) with \(a,b\in\mathbb Z\)
Page: 012
Chapter: 1
Section: 4
Kind: Proposition
Types: A Euclidean domain and a field

Q: In number theory, [...] is defined as [...].
A: \(\omega\); \((-1+\sqrt{-3})/2\)
Page: 013
Chapter: 1
Section: 4

Q: The complex \(\omega\) satisfies the equalities:
-[...],
-[...],
-[...].
A: \(\omega^3=1\); \(1+\omega+\omega^2=0\); \(\omega^2=\overline\omega\)
Page: 013
Chapter: 1
Section: 4

Q: [...] is defined as [...].
A: \(\mathbb Z[\omega]\); the set of elements of the form \(a+\omega b\) with \(a,b\in\mathbb Z\)
Page: 013
Chapter: 1
Section: 4
Index: 2
Kind: Proposition

Name: Unique factorization domain
Definition: A ring in which each nonzero is uniquely factorized into irreducibles.
Page: 013
Chapter: 1
Section: 4
Kind: Notes
Abbreviation: UFD

Q: There are [number] prime in \(\mathbb Z\).
A: infinitely many
Page: 017
Chapter: 2
Section: 1
Kind: Theo

Q: There are [...] monique irreducible polynomials in \(k[x]\).
A: infinitely many
Page: 017
Chapter: 2
Section: 1
Kind: Theo

Definition: The set of all rational numbers \(a/b\) where \(p\not| b\).
Page: 017
Chapter: 2
Section: 1
Variables: \(p\in\mathbb Z\) prime.
Notation: \(\mathbb Z_p\)
Kind: Theo

Name: Ring where all primes are associate
Page: 017
Chapter: 2
Section: 1
Example: \(\mathbb Z_p\) with \(p\) prime.

Name: A square-free integer
Definition: An integer which is not divisible by the square of any other integer greater than 1.
Page: 018
Chapter: 2

Question: \(n\) can be written in the form
Answer: \(ab^2\) where \(a,b\in\mathbb Z\) and \(a\) is square-free .
Page: 018
Chapter: 2
Variables: \(n\in\mathbb Z\).
Index: 1
Kind: Proposition

Definition: The number of positive divisors of \(n\).
Page: 018, 19
Chapter: 2
Variables: \(n\in\mathbb Z\).
Notation: \(v(n)\)
Index: , 2 a
Kind: , Prop
Definition2: \(\Pi(ord_{p_i}(n)+1)\), where \(p_i\) is the \(i\)-th prime.

Definition: The sum of positive divisors of \(n\).
Page: 018, 19
Chapter: 2
Variables: \(n\in\mathbb Z\).
Notation: \(\sigma(n)\)
Index: , 2b
Kind: , Prop
Definition2: \[\prod\frac{p_i^{a_i+1}-1}{p_i-1}\], where \(a_i\) is \(ord_{p_i}(n)\) and \(p_i\) is the \(i\)-th prime.

Name: A perfect integer
Definition: A integer \(n\) such that \(\sigma(n)=2n\).
Page: 019
Chapter: 2

Question: A Mersenne prime is
Answer: a prime of the form \(2^{m}-1\).
Page: 019
Chapter: 2

Name: The Möbius function
Definition: The function \(f\) such that \(f(1)=1\), \(f(n)=0\) if \(n\) is not square free, and \(f(p_1\dots p_l)=(-1)^l\) where the \(p_i\) are distinct positive primes.
Page: 019
Chapter: 2
Context: number theory
Notation: \(\mu\)
Index: 3
Kind: Proposition

Name: The Dirichlet product of \(f\) and \(g\)
Definition: The function \(n\mapsto\sum_{ij=n} f(i)g(j)\).
Page: 020
Chapter: 2
Variables: \(f, g\in\mathbb Z^+\to\mathbb C\).
Context: Number theory
Notation: \(f\circ g\)

Smaller0: \(f\circ g\circ h\)
Greater0: \(\Pi_{ijk=n}f(i)g(j)h(k)\)
Page: 020
Chapter: 2
Variables: \(f, g,h\in\mathbb Z^+\to\mathbb C\).
Smaller to greater: =
Hide middle: yes

Definition: \(\left\{\begin{array}{l}1\mapsto 1\\ n\mapsto 0\end{array} \right. \).
Page: 020
Chapter: 2
Context: Number theory
Notation: \(\square\)

Question: \(f\circ\square=\)
Answer: \(f\).
Page: 020
Chapter: 2
Variables: \(f\in\mathbb Z^+\to\mathbb C\).

Q: Let \(F(n)=[...]\). Then [...].
A: \sum_{d|n}f(d); \(f(n)=\sum_{d|n}\mu(d)F(n/d)=\mu\circ F\)
Page: 020
Chapter: 2
Variables: \(f\in\mathbb Z^+\to\mathbb C\).
Kind: Theo
Name: Möbius Inversion Theorem

Smaller0: \(I\circ\mu\)
Greater0: \(\square\)
Page: 020
Chapter: 2
Index: 1
Kind: Lemma
Smaller to greater: =
Hide middle: yes
Prefix Smaller: Let \(I\) be the constant function 1.

Question: \(\sum_{d|n}\phi(d)=\)
Answer: \(n\).
Page: 020
Chapter: 2
Variables: \(n\in\mathbb Z\)
Index: 4
Kind: Prop

Smaller0: \(\phi(\Pi_i p_i^{a_i} )\)
Greater0: \(\Pi_ip_i^{a_i}(1-1/p_i)=\Pi_i\frac{p_i^{a_i+1}-p_i^{a_i} }{p_i} \)
Page: 020
Chapter: 2
Variables: \((p_i)\) a finite sequence of distinct prime,
\((a_i)\in\mathbb N\) a finite sequence
Index: 5
Kind: Prop
Smaller to greater: =
Hide middle: yes

Question: The series \(\sum_{p\text{ prime} }\frac 1p\)
Answer: diverges.
Page: 21
Chapter: 2
Section: 3
Kind: Theorem

Question: The series \(\sum_{p\text{ monic irreduccible polynomial} }q^{-\text{deg}p}\)
Answer: diverges.
Page: 21
Chapter: 2
Section: 3
Variables: \(K\) a field with \(q\) elements.
Index: 4
Kind: Theorem

Question: The series \(\sum_{p\text{ monic irreduccible polynomial} }q^{-2\text{deg}p}\)
Answer: converges.
Page: 21
Chapter: 2
Section: 3
Variables: \(K\) a field with \(q\) elements.
Index: 4
Kind: Theorem

Smaller0: \(\pi(x)\)
Greater0: \( x/ \log(x) \)
Page: 25
Chapter: 2
Section: 4
Index: 4.2
Kind: Prop
Smaller to greater: \(\sim\)
Extra: 4.1, 4.2 are weaker

Definition: \(\sum_{p\le x,\ p \text{ prime} }\log p\)
Page: 23
Chapter: 2
Section: 4
Variables: \(x\ge 2\)
Notation: \(\theta(x)\)
Index: 4.3
Kind: Prop

Question: \(\theta(x)<\)
Answer: \((4\log 2)x\).
Page: 23
Chapter: 2
Section: 4
Variables: \(x\ge 2\)
Index: 4.3
Kind: Prop
Extra: trop précis

Name: Fermat prime
Definition: A prime of the form \(2^N+1\)
Page: 25
Chapter: 2
Section: 4
Index: 4.3
Kind: Prop

Page: 29
Definition: \(m\) divides \(b-a\).
Chapter: 3
Section: 2
Variables: \(a,b,m\in\mathbb Z\) with \(m\ne0\).
Notation: \(a\equiv b(m)\)
Name: \(a\) is congruent to \(b\) modulo \(m\)
Kind: Def
Equivalence: Equivalence relation

Name: The congruence class of \(a\) modulo \(m\)
Definition: \(a+m\mathbb Z\)
Page: 29
Chapter: 3
Section: 2
Variables: \(a,m\in\mathbb Z\) with \(m\ne0\).
Notation: \(\overline a\)
Kind: Def

Definition: The set of congruence classes modulo \(m\).
Page: 30
Chapter: 3
Section: 2
Variables: \(m\in\mathbb Z\) with \(m\ne0\).
Notation: \(\mathbb Z/m\mathbb Z\)
Kind: Def

Name: A complete set of residues modulo \(m\)
Definition: A set containing exactly one element from each congruence classes.
Page: 30
Chapter: 3
Section: 2
Variables: \(m\in\mathbb Z\) with \(m\ne0\).
Kind: Def

Smaller0: \(\overline a+\overline b\)
Greater0: \(\overline{a+b} \)
Page: 30
Chapter: 3
Section: 2
Variables: \(a,b,m\in\mathbb Z\) with \(m\ne0\).
Kind: Def
Smaller to greater: =
Hide middle: yes

Smaller0: \(\overline a\overline b\)
Greater0: \(\overline{ab} \)
Page: 30
Chapter: 3
Section: 2
Variables: \(a,b,m\in\mathbb Z\) with \(m\ne0\).
Kind: Prop
Smaller to greater: =
Hide middle: yes

Smaller0: \(p(0)\) and \(p(1)\) are both odds
Greater0: \(p(x)\ne 0\) for \(x\in\mathbb Z\)
Page: 31
Chapter: 3
Section: 2
Variables: \(p\in\mathbb Z[x]\)

Question: The set of solutions of \(ax\equiv b(m)\) is:
-
Answer: let \(d=\text{gcd}(a,m)\). If \(d|b\) then \(\{x_0+im/d\mid i\in[0,d-1]\}\) for some solution \(x_0\).
Otherwise there is no solution
Page: 32
Chapter: 3
Section: 2
Variables: \(a,b,m\in\mathbb Z\), \(m\ne 0\).
Index: 1
Kind: Prop

Smaller0: \(a,m\) are relatively prime
Greater0: exactly one solution
Page: 32
Chapter: 3
Section: 2
Variables: \(a,b,m\in\mathbb Z\)
\(m\ne 0\)
Index: 1,2
Kind: Cor
Prefix Greater: \(ax\equiv b(m)\) have

Smaller0: a unit of \(\mathbb Z/m\mathbb Z\)
Greater0: \((a,m)=1\)
Page: 33
Chapter: 3
Section: 2
Variables: \(a,m\in\mathbb Z\), \(m\ne 0\)
Kind: Prop
Smaller to greater: iff
Hide middle: yes
Prefix Smaller: \(\overline a\) is

Question: The number of unit of \(\mathbb Z/m\mathbb Z\) is
Answer: \(\phi(m)\).
Page: 33
Chapter: 3
Section: 2
Variables: \(a,m\in\mathbb Z\), \(m\ne 0\)
Kind: Prop

Smaller0: \((a,m)=1\)
Greater0: \(a^{\phi(m)}\equiv 1(m)\)
Page: 33
Chapter: 3
Section: 2
Variables: \(a,m\in\mathbb Z\)
\(m\ne 0\)
Index: 1
Kind: Cor
Name: Euler's Theorem

Smaller0: \(p\) is prime and \(p\) does not divide \(a\)
Greater0: \(a^{p-1}\equiv 1(p)\)
Page: 33
Chapter: 3
Section: 2
Variables: \(a,p\in\mathbb Z\)
Kind: Cor
Hide middle: yes
Name: Fermat's Little Theorem
Suffix Greater: ,

Smaller0: the \(a_i\) are relatively prime to \(m\)
Greater0: \(\Pi_{i=1}^na_i\) is relatively prime to \(m\)
Page: 34
Chapter: 3
Section: 4
Variables: \(a_1,\dots,a_n,m\in\mathbb Z\).
Index: 1
Kind: Lem
Hide middle: yes

Smaller0: the \(a_i\) divide \(m\) and \((a_i,a_j)=1\) for \(i\ne j\)
Greater0: \(\Pi_{i=1}^na_i\) divides \(m\)
Page: 34
Chapter: 3
Section: 4
Variables: \(a_1,\dots,a_n,m\in\mathbb Z\).
Index: 2
Kind: Lem
Hide middle: yes

Definition: The set of units of \(R\).
Page: 35
Chapter: 3
Section: 4
Variables: \(R\) a ring.
Notation: \(U(R)\)

Page: 35
Chapter: 3
Section: 4
Name: The direct sum of the \(R_i\)s
Set: \(\prod R_i\)
Variables: \(R_1,\dots,R_n\) rings.
Notation: \(R_1\oplus \dots\oplus R_n\)
*: Componentwise
0: Tuple of 0
1: Tuple of 1
Extra: It is indeed sum, and not product

Question: \(U(R_1\oplus\dots\oplus R_n)=\)
Answer: \(U(R_1)\times \dots\times U(R_n)\).
Page: 35
Chapter: 3
Section: 4
Variables: \(R_1,\dots,R_n\) rings.
Index: 4.1
Kind: Prop

\(m_1,\dots,m_t\) are pairwise relatively prime implies \(\mathbb Z/\left(\Pi_{i=1}^tm_i\right) \mathbb Z\) is isomorphic to \(\left(\mathbb Z/m_1\mathbb Z\right)\oplus\dots\oplus\left(\mathbb Z/m_t\mathbb Z\right)\).
Page: 35
Chapter: 3
Section: 4
Variables: \(m_1,\dots,m_t\in\mathbb Z\).
Index: 1'
Kind: Theo

Question: The number of roots of \(f\) is
Answer: at most \(n\).
Page: 39
Chapter: 4
Section: 1
Variables: \(k\) a field and \(f(x)\in k[x]\) of degree \(n\).
Kind: Lem

Smaller0: at least \(n+1\) elements
Greater0: \(f=g\)
Page: 39
Chapter: 4
Section: 1
Variables: \(k\) a field and \(f(x),g(x)\in k[x]\) of degree \(n\).
Kind: Cor
Hide middle: yes
Prefix Smaller: \(g(x)=f(x)\) on

Smaller0: \(n\) prime
Greater0: \(\Pi_{i=1}^{n-1}(x-i)\)
Page: 40
Chapter: 4
Section: 1
Variables: \(n\in\mathbb N\).
Index: 1.1
Kind: Prop
Hide middle: yes
Greater1: \(x^{n-1} \)
Connect Greater0: \(\equiv\)
Connect Greater1: \(\mod m\)

Q: \((n-1)!\equiv [...]\ (n)\)
A: \left\{\begin{array}{ll}-1&\text{if $n$ is prime}\\2&\text{if $n=4$}\\0&\text{otherwise}\end{array}\right.
Page: 40
Chapter: 4
Section: 1
Variables: \(n\in\mathbb N\).
Index: Wilson's theorem
Kind: Cor

Name: A primitive root mod \(n\)
Definition: \(\overline a\) generates the group \(U(\mathbb Z/n\mathbb Z)\)
Page: 41
Chapter: 4
Section: 1
Variables: \(a,n\in\mathbb Z\)
Kind: Def
Definition2: An integer \(a\) such that \(\phi(n)\) is the smallest positive integer such that \(a^{\phi(n)}\equiv 1(n)\)

Smaller0: \(b\) is prime
Greater0: \(0\)
Page: 42
Chapter: 4
Section: 1
Variables: \(0<a<b\in\mathbb N\)
Index: 2
Kind: Lem
Hide middle: yes
Prefix Greater: \({b\choose a}\equiv \)
Greater1: \(b\)
Connect Greater0: \(\,(\)
Suffix Greater: \()\)

Smaller0: \(a\equiv b (n^l)\)
Greater0: \(n^{l+1} \)
Page: 42
Chapter: 4
Section: 1
Variables: \(a,b,n\in\mathbb Z\)
\(l>0\)
Index: 3
Kind: Lem
Hide middle: yes
Prefix Greater: \(a^n\equiv b^n (\)
Assuming: \(n\) is prime
Suffix Greater: \()\)

Q: \([...]\equiv [...]\,([...])\)
A: (1+an)^{n^{l-2} }; 1+an^{l-1}; n^l
Page: 42
Chapter: 4
Section: 1
Variables: \(a,n\in\mathbb N\)
\(l>1\)
Kind: Cor
Assuming: \(n\) is an odd prime

Name: The order of \(a\) modulo \(n\)
Definition: The smallest integer \(e\) such that \(a^e\equiv 1\ (n)\).
Page: 43
Chapter: 4
Section: 1
Variables: \(a,n\in\mathbb Z\)
Kind: Def

\(n\) does not divide \(a\) implies the order of \(1+an\) modulo \(n^l\) is \(1+ap\).
Page: 43
Chapter: 4
Section: 1
Variables: \(a,n\in\mathbb Z\)
Index: 2
Kind: Cor
Extra: l?
Assuming: \(n\) is an odd prime

Question: \(U(\mathbb Z/n\mathbb Z)\) is isomorphic to
Answer: \(\Pi U(\mathbb Z/p_i^{a_i}\mathbb Z)\).
Page: 44
Chapter: 4
Section: 1
Variables: \(n=\Pi p_i^{a_i}\)
Index: 3
Kind: Theo

Smaller0: \(n\) possesses primitive roots
Greater0: \(n\) is 2, 4, \(p^a\) or \(2p^a\) with \(p\) an odd prime and \(a\in\mathbb N\).
Equivalently: \(n\in\{1,2\}\{p^a\mid p\text{ prime}, p>2\lor a=1\} \)
Page: 44
Chapter: 4
Section: 1
Variables: \(n\in\mathbb N\)
Index: 1.3
Kind: Prop
Smaller to greater: iff
Hide middle: yes
Extra: Implies Theo 2, 2'

Name: An \(n\)th power residue \(\mod m\)
Definition: An integer \(a\) such that \((a,m)=1\), \(x^n\equiv a\,(m)\) is solvable.
Page: 45
Chapter: 4
Section: 2
Variables: \(m,n>0\), \(a\in\mathbb Z\).
Kind: Def

Smaller0: \((a,m)=1\) and \(m\) posesses primitive roots
Greater0: \(a^{\phi(m)/(n,\phi(m))}\equiv 1\,(m)\)
Page: 45
Chapter: 4
Section: 2
Variables: \(m,n>0\), \(,a\in\mathbb Z\).
Index: 1
Kind: Prop
Smaller to greater: iff
Smaller1: an \(n\)th power residue \(\mod m\)
Prefix Smaller: If
Connect Smaller0: then \(a\) is

Smaller0: \(e\ge3\), \(a\) odd, and \(n\) odd
Greater0: exactly 1
Page: 45
Chapter: 4
Section: 2
Variables: \(n,e>0\), \(a\in\mathbb Z\).
Index: 3
Kind: Prop
Hide middle: yes
Prefix Greater: \(x^n\equiv a\,(2^e)\) has
Suffix Greater: solution(s).

Question: The number of solutions of \(x^n\equiv a\,(2^e)\) is
Answer: \(2(n,e^{e-2})\) if \(a\equiv 1\,(4)\) and \(a^{2^{e-2}/(n,2^{e-2})}\equiv 1\,(2^e)\) and 0 otherwise.
Page: 45
Chapter: 4
Section: 2
Variables: \(n,e>0\), \(a\in\mathbb Z\).
Index: 3
Kind: Prop
Assuming: \(a\) odd, \(e\ge3\), \(n\) even

Smaller0: \(m\) is an odd prime, not dividing \(a\) nor \(n\)
Greater0: at least \(1\)
Page: 45
Chapter: 4
Section: 2
Variables: \(m,n,e>0\), \(a\in\mathbb Z\).
Index: 3
Kind: Prop
Hide middle: yes
Prefix Greater: \(x^n\equiv a\,(m)\) has
Suffix Greater: solution(s)

Smaller0: \(m\) is an odd prime, not dividing \(a\) nor \(n\)
Greater0: the same number of solutions than \(x^n\equiv a\, (m)\)
Page: 46
Chapter: 4
Section: 2
Variables: \(m,n,e>0\), \(a\in\mathbb Z\).
Index: 3
Kind: Prop
Hide middle: yes
Prefix Greater: \(x^n\equiv a\,(m^e)\) has

Smaller0: \(a\) is an odd prime
Greater0: the same number of solutions than \(x^n\equiv a\, (2^{2l+1})\), at least one
Page: 46
Chapter: 4
Section: 2
Variables: \(n,e>0\), \(a,l\in\mathbb Z\).
Kind: Prop
Prefix Greater: \(x^n\equiv a\,(2^e)\) has
Smaller1: \(l\) maximal such that \(2^l\) divides \(n\) and \(e\ge 2l+1\)

Name: A quadratic residue \(\mod m\)
Definition: An integer \(a\), coprime with \(m\), such that \(x^2\equiv a\mod m\) has a solution.
Page: 50
Chapter: 5
Section: 1
Variables: \(m\in\mathbb Z\)

Smaller0: \(a\) is a quadratic residue \(\mod m\)
Greater0: \(e=2\) implies \(a\equiv 1\,(4)\)
Page: 50
Chapter: 5
Section: 1
Variables: \(m=2^e\Pi p_i^{e_i}\) and \(a\in\mathbb Z\)
Kind: Prop
Smaller to greater: iff
Assuming: \((a,m)=1\)
Greater1: \(e\ge3\) implies \(a\equiv 1\,(8)\) and
Extra: i ?
Greater2: \(a^{(i-1)/2}\equiv 1\,(p_i)\)

Name: \(\mathbb Z[\omega]\)
Page: 013
Chapter: 1
Section: 4
Index: 2
Kind: Prop
Typ: A Euclidean domain and a field

\(a^{(n-1)/2} \equiv (a/n)\mod n\).
Page: 51
Chapter: 5
Section: 1
Variables: \(a,n\in\mathbb Z\)
Index: 2 (a)
Kind: Prop
Assuming: \(n\) an odd prime

Smaller0: \(a\equiv b \mod n\)
Greater0: \((a/n)=(b/n)\)
Page: 51
Chapter: 5
Section: 1
Variables: \(a,b,n\in\mathbb Z\)
Index: 2 (c)
Kind: Prop
Assuming: \(n\) an odd prime

Question: The number of non-quadratic residues modulo \(p\) is
Answer: equal to the number of quadratic residues modulo \(p\)
Page: 51
Chapter: 5
Section: 1
Variables: \(p\) an odd prime
Kind: Cor

Question: The product of a quadratic residue and of a quadratic residue is
Answer: a quadratic residue.
Page: 51
Chapter: 5
Section: 1
Variables: \(p\) an odd prime
Index: 2
Kind: Cor

Question: The product of a quadratic nonresidue and of a quadratic nonresidue is
Answer: a quadratic residue.
Page: 51
Chapter: 5
Section: 1
Variables: \(p\) an odd prime
Index: 2
Kind: Cor

Question: The product of a quadratic residue and of a quadratic nonresidue is
Answer: a quadratic nonresidue.
Page: 51
Chapter: 5
Section: 1
Variables: \(p\) an odd prime
Index: 2
Kind: Cor

Q: \([...][...][...]\).
A: (-1)^{(p-1)/2}; =; (-1/p)
Page: 52
Chapter: 5
Section: 1
Variables: \(p\) an odd prime
Index: 3
Kind: Cor

Question: The number of prime congruent to \(1\) mod \(4\) is
Answer: infinite.
Page: 52
Chapter: 5
Section: 1
Index: 3
Kind: Cor

Name: The set of least residues mod \(p\)
Definition: \([-(p-1)/2,\dots,(p-1)/2]\).
Page: 52
Chapter: 5
Section: 1
Variables: \(p\) an odd prime
Index: 3
Kind: Cor

Question: \((a/p)\) equals
Answer: \((-1)^\mu\) where \(\mu\) is the number of elements in \(a,2a,\dots,((p-1)/2)a\) whose least residue is negative.
Page: 52
Chapter: 5
Section: 1
Variables: \(p\) an odd prime,
\(a\) not a multiple of \(p\)
Kind: Lem

Smaller0: \(2\) is a quadratic residue mod \(p\)
Greater0: \(p\) is equivalent to \(\pm 1\) mod \(8\)
Page: 53
Chapter: 5
Section: 1
Variables: \(p\) an odd prime,
Index: 3
Kind: Prop
Smaller to greater: iff

Question: The number of solution of \(x^n\equiv a(m)\) is
Answer: 0 or \((n,\phi(m))\).
Page: 45
Chapter: 4
Section: 2
Variables: \(m,n>0\), \(a\in\mathbb Z\).
Index: 1
Kind: Prop

Definition: \((d)\) where \(d\) is a greatest commun divisor of \(a\) and \(b\).
Page: 004
Chapter: 1
Variables: \(a,b\in\mathbb Z\).
Notation: \((a,b)\)
Index: 3, 4
Kind: Lemma

Name: A greatest common divisor of \(f\) and \(g\)
Definition: An integer \(d\) which is a divisor of \(f\) and of \(g\) and if every common divisor of \(f\) and of \(g\) are divisor of \(d\)
Page: 007
Chapter: 1
Section: 2
Variables: \(f,g\in k[x]\).
Kind: Def

Smaller0: \((m_i,m_j)=(1)\)
Greater0: the set of solutions of \(\left(x\equiv b_i (m_i)\right)_{i=1}^t\), with \(b_1,\dots,b_t\in\mathbb Z\) is \(y+m\mathbb Z\) for some \(y\in\mathbb Z\)
Page: 34
Chapter: 3
Section: 4
Variables: \(t\in\mathbb N\)
\(m_1,\dots, m_t\in\mathbb Z\).
Index: 1
Kind: Theo
Name: Chinese Remainder Theorem

Smaller0: \((ab/n) \)
Greater0: \((a/n)(b/n)\)
Page: 51
Chapter: 5
Section: 1
Variables: \(a,b,n\in\mathbb Z\)
Index: 2 (b)
Kind: Prop
Smaller to greater: =
Assuming: \(n\) an odd prime

Smaller0: \(-1\) is a square residue
Greater0: \(p\equiv -1 (4)\)
Page: 52
Chapter: 5
Section: 1
Variables: \(p\) an odd prime
Index: 3
Kind: Cor
Smaller to greater: iff

Smaller0: \(R\) is a Euclidean domain
Greater0: \(X\) is a pid
Page: 009
Chapter: 1
Section: 3
Variables: \(R\) an integral domain
\(X\subseteq R\).
Kind: Proposition
Hide middle: yes
Smaller1: \(X\) is an ideal

Smaller0: \(F\) is a PID
Greater0: \(F\) is a UFD
Page: 013
Chapter: 1
Section: 4
Variables: \(F\) a field
Kind: Notes
Counterexample: The polymonials over a field in more than one variables

Smaller0: \(n\) is prime
Greater0: \(d\)
Page: 40
Chapter: 4
Section: 1
Variables: \(d,n\in\mathbb N\)
Index: 1.2
Kind: Prop
Prefix Greater: the number of solutions of \(x^d\equiv 1(n)\) is
Smaller1: \(d|n-1\)

Name: The Legendre symbol
Definition: 1 if \(a\) is a quadratic residue mod \(n\)
-1, if \(a\) is a quadratic nonresidue mod \(n\)
0 if \(n|a\).
Page: 51
Chapter: 5
Section: 1
Notation: \((a/n)\)
Assuming: \(n\) an odd prime,
\(a\in\mathbb Z\)

Smaller0: a field
Greater0: \(m\) is prime
Page: 33
Chapter: 3
Section: 2
Variables: \(R\) a pid
\(m\in R\)
Smaller to greater: iff
Hide middle: yes
Prefix Smaller: \(R/(m)\) is

Name: A quadratic nonresidue \(\mod m\)
Definition: An integer \(a\), coprime with \(a\), such that \(x^2\equiv a\mod m\) has no solution
Page: 50
Chapter: 5
Section: 1
Variables: \(m\in\mathbb Z\)