Fields, local objects, and orders catalogue - Knowlpedia

Core idea

69 separately identified objects and parameterized families. Real and complex scalar choices have distinct entries. An isomorphism is a recorded relationship, not a reason to collapse the entries.

Explore pairs and categories · Catalogue overview

Adele ring

Object	Dimensions	Kind of entry
$\mathbb A_{\mathbb F_q(t)}$	not specified	parameterized family
$\mathbb A_{K}$	not specified	parameterized family
$\mathbb A_{K}$	not specified	parameterized family
$\mathbb A_{K,f}$	not specified	parameterized family
$\mathbb A_{\mathbb Q}$	not specified	specified object
$\mathbb A_{\mathbb Q,f}$	not specified	specified object

Finite field

Object	Dimensions	Kind of entry
$\mathbb F_{27}$	prime_field: 3; F3: 3	specified object
$\mathbb F_4$	prime_field: 2; F2: 2	specified object
$\mathbb F_8$	prime_field: 3; F2: 3	specified object
$\mathbb F_9$	prime_field: 2; F3: 2	specified object
$\mathbb F_{p^2}$	prime_field: 2; Fp: 2	parameterized family
$\mathbb F_{p^3}$	prime_field: 3; Fp: 3	parameterized family
$\mathbb F_{p^k}$	prime_field: k; Fp: k	parameterized family

Idele group

Object	Dimensions	Kind of entry
$\mathbb A_{\mathbb F_q(t)}^\times$	not specified	parameterized family
$\mathbb A_{K}^\times$	not specified	parameterized family
$\mathbb A_{K}^\times$	not specified	parameterized family
$\mathbb A_{K,f}^\times$	not specified	parameterized family
$\mathbb A_{\mathbb Q}^\times$	not specified	specified object
$\mathbb A_{\mathbb Q,f}^\times$	not specified	specified object

Integer ring

Object	Dimensions	Kind of entry
$\mathbb Z$	not specified	specified object

Laurent series field

Object	Dimensions	Kind of entry
$\mathbb F_2((t))$	not specified	specified object
$\mathbb F_3((t))$	not specified	specified object
$\mathbb F_5((t))$	not specified	specified object
$\mathbb F_q((t))$	not specified	parameterized family

Matrix algebra

Object	Dimensions	Kind of entry
$M_1(\mathbb Q)$	rational: 1	specified object
$M_2(\mathbb Q)$	rational: 4	specified object
$M_3(\mathbb Q)$	rational: 9	specified object
$M_n(\mathbb Q)$	rational: n^2	parameterized family

Padic field

Object	Dimensions	Kind of entry
$\mathbb Q_2$	self: 1	specified object
$\mathbb Q_3$	self: 1	specified object
$\mathbb Q_5$	self: 1	specified object
$\mathbb Q_p$	self: 1	parameterized family

Padic integer ring

Object	Dimensions	Kind of entry
$\mathbb Z_2$	not specified	specified object
$\mathbb Z_3$	not specified	specified object
$\mathbb Z_5$	not specified	specified object
$\mathbb Z_p$	not specified	parameterized family

Polynomial ring

Object	Dimensions	Kind of entry
$\mathbb F_2[t]$	not specified	specified object
$\mathbb F_3[t]$	not specified	specified object
$\mathbb F_5[t]$	not specified	specified object
$\mathbb F_q[t]$	not specified	parameterized family

Power series ring

Object	Dimensions	Kind of entry
$\mathbb F_2[\![t]\!]$	not specified	specified object
$\mathbb F_3[\![t]\!]$	not specified	specified object
$\mathbb F_5[\![t]\!]$	not specified	specified object
$\mathbb F_q[\![t]\!]$	not specified	parameterized family

Prime field

Object	Dimensions	Kind of entry
$\mathbb F_2$	prime_field: 1; F2: 1	specified object
$\mathbb F_3$	prime_field: 1; F3: 1	specified object
$\mathbb F_5$	prime_field: 1	specified object
$\mathbb F_p$	prime_field: 1; Fp: 1	parameterized family

Quadratic integer ring

Object	Dimensions	Kind of entry
$\mathbb Z[\omega]$	not specified	specified object
$\mathbb Z[i]$	not specified	specified object

Quadratic number field

Object	Dimensions	Kind of entry
$\mathbb Q(i)$	rational: 2	specified object
$\mathbb Q(\sqrt{-3})$	rational: 2	specified object
$\mathbb Q(\sqrt2)$	rational: 2	specified object
$\mathbb Q(\sqrt d)$	rational: 2	parameterized family

Quaternion algebra

Object	Dimensions	Kind of entry
$(-1,-1)_{\mathbb Q}$	rational: 4	specified object
$(a,b)_F$	F: 4	parameterized family

Quaternion order

Object	Dimensions	Kind of entry
$\mathcal O_{\mathrm{Hur}}$	not specified	specified object
$\mathbb Z\langle i,j\rangle$	not specified	specified object

Rational field

Object	Dimensions	Kind of entry
$\mathbb Q$	rational: 1	specified object

Rational function field

Object	Dimensions	Kind of entry
$\mathbb F_2(t)$	not specified	specified object
$\mathbb F_3(t)$	not specified	specified object
$\mathbb F_5(t)$	not specified	specified object
$\mathbb F_q(t)$	not specified	parameterized family

Symbolic algebra over extension

Object	Dimensions	Kind of entry
$A/K$	K: dim_K(A); F: [K:F] dim_K(A)	parameterized family

Symbolic base field

Object	Dimensions	Kind of entry
$F$	F: 1	parameterized family

Symbolic field extension

Object	Dimensions	Kind of entry
$K/F$	K: 1; F: [K:F]	parameterized family

Unramified padic extension

Object	Dimensions	Kind of entry
$\mathbb Q_p^{(2),\mathrm{ur}}$	Q_p: 2	parameterized family
$\mathbb Q_p^{(3),\mathrm{ur}}$	Q_p: 3	parameterized family
$\mathbb Q_p^{(n),\mathrm{ur}}$	Q_p: n	parameterized family