Documentation

Creation Functions

Creation of Structures

The ring of integers is automatically created when Magma is first loaded. The ring may be formally created (and, if desired, assigned to a variable) using the function IntegerRing(). Subrings of Z are always ideals; see the section on ideals for details.

IntegerRing() : -> RngInt

Integers() : -> RngInt

RingOfIntegers() : -> RngInt

IntegerRing(Q) : FldRat -> RngInt

Integers(Q) : FldRat -> RngInt

RingOfIntegers(Q) : FldRat -> RngInt

Create the ring of integers Z.

Creation of Elements

Since the ring of integers is present when Magma is started up, integers typed into Magma without any explicit context will be regarded as elements of the ring of integers. Integers can be specified using both decimal and hexadecimal notation.

One(Z) : RngInt -> RngIntElt

Identity(Z) : RngInt -> RngIntElt

Zero(Z) : RngInt -> RngIntElt

Representative(Z) : RngInt -> RngIntElt

These generic functions create 1, 1, 0, and 0 respectively, in the integer ring Z.

a₁a₂...a_r

Given a succession of decimal digits a₁, ..., a_r, create the corresponding integer. Leading zeros will be ignored.

0xa₁a₂...a_r

Given a succession of hexadecimal digits a₁, ..., a_r, create the corresponding integer. Leading zeros will be ignored.

elt< Z | a₁a₂...a_r > : RngInt, RngIntElt -> RngIntElt

Given a succession of decimal digits a₁, ..., a_r, create the corresponding integer as an element of Z.

elt< Z | 0xa₁a₂...a_r > : RngInt, RngIntElt -> RngIntElt

Given a succession of hexadecimal digits a₁, ..., a_r, create the corresponding integer as an element of Z.

Z ! a : RngInt, RngElt -> RngIntElt

Z ! [a] : RngInt, [RngElt] -> RngIntElt

Coerce the ring element a into the ring of integers Z. The element a is allowed to be an element of the ring of integers modulo m (in which case the result r satisfies 0 ≤r < m), or an element of a finite field (in which case the result r satisfies 0 ≤r < p if a is in the prime field, of characteristic p, and an error otherwise), or an element of the integers, rationals, a quadratic field, a cyclotomic field or a number field (in which cases the result is the obvious integer if a is integral and an error otherwise).

Example `RngInt_Integers (H19E2)`

> Z := IntegerRing();
> n := 1234567890;
> n in Z;
true
> m := elt< Z | 1234567890 >;
> m eq n;
true
> l := Z ! elt< QuadraticField(3) | 1234567890, 0>;
> l;
1234567890
> k := elt< Z | 0x499602D2 >;
1234567890

Printing of Elements

Magma supports the printing of integers in both decimal and hexadecimal form. The default print method is to print integers in base 10; base 16 printing is performed using the Hex print level.

Example `RngInt_Printing (H19E3)`

> n := 1234567890;
> n;
1234567890
> n:Hex;
0x499602D2

Element Conversions

FactorizationToInteger(s) : [ <RngIntElt, RngIntElt> ] -> RngIntElt

FactorisationToInteger(s) : [ <RngIntElt, RngIntElt> ] -> RngIntElt

Facint(s) : [ <RngIntElt, RngIntElt> ] -> RngIntElt

Given a sequence of two-element tuples s=[ <p₁, k₁>, ..., <p_r, k_r> ] containing pairs of integers <p_i, k_i>, 1 ≤i ≤r, with k_i non-negative, this function returns the integer p₁^k₁ ... p_r^k_r. It is normally used for converting a factorization sequence to the corresponding integer.

IntegerToSequence(n, b) : RngIntElt, RngIntElt -> [RngIntElt]

Intseq(n, b) : RngIntElt, RngIntElt -> [RngIntElt]

Given a non-negative integer n and a positive integer b≥2, return the unique base b representation of n in the form of a sequence Q. That is, if n = a₀b⁰ + a₁b¹ + ... + a_{k - 1}b^{k - 1} with 0≤a_i<b and a_{k - 1}> 0, then Q = [ a₀, a₁, ..., a_{k - 1} ]. (If n=0, then Q=[ ].)

SequenceToInteger(s, b) : [RngIntElt], RngIntElt -> RngIntElt

Seqint(s, b) : [RngIntElt], RngIntElt -> RngIntElt

Given a positive integer b≥2 and a sequence Q = [ a₀, ..., a_{k - 1} ] of non-negative integers such that 0 ≤a_i < b, return the integer n = a₀b⁰ + a₁b¹ + ... + a_{k - 1}b^{k - 1}. If Q is the empty sequence, the integer zero is returned. This function performs the inverse operation of the base b representation.

IntegerToString(n) : RngIntElt -> ModStgElt

Create the string consisting of the decimal digits of the integer n. In the case in which n is negative the first character will be the minus sign.

IntegerToString(n, b) : RngIntElt, RngIntElt -> ModStgElt

Create the string consisting of the digits of the integer n in base b. In the case in which n is negative the first character will be the minus sign. The base b can be between 2 and 36. For b ≤10, the digits are represented numerically. For b > 10, the digits are represented both numerically and alphabetically, so that, 10 is `A', 11 is `B', et cetera.

Eltseq(n) : RngIntElt -> [RngIntElt]

The sequence [n] which can be coerced back into Z.

Denominator(n) : RngIntElt -> RngIntElt

The denominator of n, ie. 1.

V2.28, 13 July 2023

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