For every prime $p$, the two operations on $\mathbb{Z}\mathrm{/}p$ make it a field

Depends on

• For every natural $n$, $(\mathbb{Z}/n,+)$ is an abelian group, multiplication is a commutative monoid operation, and both distributive laws hold
• For $n\ge1$, $[a]_n$ is a unit if and only if $\gcd(a,n)=1$
• For a prime $p$ and any integer $a$, $\gcd(p,a)$ is $p$ when $p \mid a$ and $1$ otherwise; so $p \nmid a$ makes $p$ and $a$ coprime
• Prime and composite integers: $p$ is prime when $p > 1$ and its only positive divisors are $1$ and $p$
• Field
• The congruence class $[a]_n$ and the quotient set $\mathbb{Z}/n$
• Congruence modulo an integer: $a\equiv b\pmod n$ when $n\mid(a-b)$, including the moduli $0$ and $1$

Used by

• For an odd prime p, ℚ(ζₚ) has exactly one intermediate field of degree two over ℚ Corollary
• For every prime p, the multiplicative group (ℤ/pℤ)^× is cyclic Corollary
• x²≡ a (mod p) has exactly 1+(a/p) solution classes Corollary
• A set family whose incidence vectors are dependent over F₂ and independent over ℝ Counterexample
• In characteristic 2, a symmetric bilinear form need not have an orthogonal basis Counterexample
• In characteristic 2, distinct quadratic forms can have the same polar form Counterexample
• In ℤ/4 the sets A=B={0,2} have | A+B|=2, below the Cauchy–Davenport bound 3 Counterexample
• Over F₂, the equation x+y=0 has exactly two solutions, so the infinite-field hypothesis is necessary Counterexample
• Over Fₚ, xᵖ-x and the zero polynomial induce the same function but are distinct polynomials Counterexample
• Over ℤ/2, an antisymmetric bilinear form need not be alternating Counterexample
• Power sums need not generate the symmetric ring in characteristic two Counterexample
• The commutator pairing of an extraspecial p-group relative to a chosen generator of its centre Definition
• The Heisenberg group of order p³ over ℤ/p Definition
• The square map of an extraspecial 2-group relative to a chosen generator of its centre Definition
• A dual basis and transpose computed for a linear map on F₃² Example
• A localization of the integers at p need not be Henselian Example
• A nonmonic quadratic congruence solved through its discriminant Example
• beginpmatrix0&11&1 endpmatrix over F₂ has characteristic polynomial x²+x+1 and no eigenvalue in its base field Example
• F₄=F₂[t]/(t²+t+1) with complete addition and multiplication tables Example
• F₈=F₂[t]/(t³+t+1) and its power table Example
• F₉=F₃[t]/(t²+1) and a generator of its multiplicative group Example
• For every integer n>1, nℤ is a maximal ideal of ℤ if and only if n is prime Example
• Frobenius on F₄ swaps the two non-prime-field elements Example
• In characteristic 2, x²+1=(x+1)² has zero derivative and a repeated root Example
• Over F₂, x⁴+x²+1=(x²+x+1)² has two distinct roots, each repeated, in its four-element splitting field Example
• Pairwise independent affine hashing over a prime field Example
• Reduction modulo 2 proves x³+17x+391 irreducible over ℚ Example
• Schwartz zippel for a bivariate polynomial Example
• Spec Z has one generic point and closed prime-number points Example
• Sylow p-subgroups of Aut((ℤ/p)²): nₚ=p+1 Example
• Sylow subgroups of Aff(ℤ/5): n₂=5 and n₅=1 Example
• The 3×3 upper-unitriangular group over a prime field has generator rank two Example
• The affine group AGL(1,p) has one kernel-conjugacy class of complements to its translation subgroup Example
• The finite Heisenberg group is the unique Sylow p-subgroup of its coordinate upper-triangular group Example
• The four-element field (ℤ/2)[x]/(x²+x+1) Example
• The Frattini subgroup of (ℤ/p)ⁿ is trivial Example
• The prime ideals of a field and of the integers Example
• The residue class of t generates F₈^× Example
• ℤ₍ₚ₎ consists of rationals with denominator not divisible by p, has maximal ideal pℤ₍ₚ₎, and residue field Fₚ Example
• FALSE for odd p: the scalar-valued commutator pairing needs no choice of a central generator False statement

…and 31 more results.

Dependency tree · two levels

34 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• K. Conrad, Modular Arithmetic (standard reference, not scraped)
• MIT 18.783 Lecture Notes 3 (standard reference, not scraped)