Introduction

OSCAR provides functionality for working with a wide variety of different fields. The methods applicable to any field and its element can be found in the interface section. For functionality available only for specific fields, consult the corresponding section of the manual.

Available fields

Here is a list of the fields available in OSCAR:

Field	How to create	Remark	Reference
$\mathbb{Q}$	`rational_field()`	Also available as `QQ`	Rationals
$\mathbb{F}_q$	`GF(q)`	See also `finite_field`	Finite fields
$\mathbb{F}_q[X]/(f)$	`finite_field(f)`		Finite fields
$\overline{\mathbb{Q}}$	`algebraic_closure(QQ)`		Algebraic closure of the rational numbers
$\overline{\mathbb{F}}_q$	`algebraic_closure(F)`		Algebraic closure of finite prime fields
$\mathbb{Q}^{\mathrm{ab}}$	`abelian_closure(QQ)`		Abelian closure of the rationals
$\mathbb{Q}[X]/(f)$	`number_field(f)`
$\mathbb{Q}(\alpha) \subseteq \R$	`embedded_number_field`	Ordered field
$\mathbb{R}$	`real_field()`	Ball arithmetic	Arbitrary precision real balls
$\mathbb{C}$	`complex_field()`	Ball arithmetic	Arbitrary precision complex balls
$\mathbb{Q}_p$	`padic_field(p)`		Padics
$\mathbb{Q}_{p^n}$	`qadic_field(p, n)`	Unramified extensions of $\mathbb{Q}_p$	Qadics
$R/(f)$	`residue_field(R, f)`	$R$ must be a principal ideal domain
$\mathrm{Quot}(R)$	`fraction_field(R)`	$R$ must be an integral domain	Generic fraction fields

Converting between fields

For fields $K$ and $L$ that admit a "canonical" embedding $K \to L$, elements from $K$ can be converted to elements from $L$ using "coercion" as in the following example:

julia> a = QQ(2)
2

julia> Qbar = algebraic_closure(QQ);

julia> b = Qbar(a)
{a1: 2.00000}

julia> C = complex_field();

julia> sqrt(C(b))
[1.414213562373095049 +/- 3.45e-19]

julia> A = QQ[1 2; 3 4]
[1   2]
[3   4]

julia> B = change_base_ring(C, A)
[1.0000000000000000000   2.0000000000000000000]
[3.0000000000000000000   4.0000000000000000000]

Contact

Please direct questions about this part of OSCAR to the following people:

You can ask questions in the OSCAR Slack.

Alternatively, you can raise an issue on github.