# Formal sums¶

AUTHORS:

• David Harvey (2006-09-20): changed FormalSum not to derive from “list” anymore, because that breaks new Element interface
• Nick Alexander (2006-12-06): added test cases.
• William Stein (2006, 2009): wrote the first version in 2006, documented it in 2009.
• Volker Braun (2010-07-19): new-style coercions, documentation added. FormalSums now derives from UniqueRepresentation.
FUNCTIONS:
• FormalSums(ring) – create the module of formal finite sums with
coefficients in the given ring.
• FormalSum(list of pairs (coeff, number)) – create a formal sum

EXAMPLES:

sage: A = FormalSum([(1, 2/3)]); A
2/3
sage: B = FormalSum([(3, 1/5)]); B
3*1/5
sage: -B
-3*1/5
sage: A + B
2/3 + 3*1/5
sage: A - B
2/3 - 3*1/5
sage: B*3
9*1/5
sage: 2*A
2*2/3
sage: list(2*A + A)
[(3, 2/3)]

class sage.structure.formal_sum.FormalSum(x, parent=None, check=True, reduce=True)

A formal sum over a ring.

reduce()

EXAMPLES:

sage: a = FormalSum([(-2,3), (2,3)], reduce=False); a
-2*3 + 2*3
sage: a.reduce()
sage: a
0

class sage.structure.formal_sum.FormalSums

The R-module of finite formal sums with coefficients in some ring R.

EXAMPLES:

sage: FormalSums()
Abelian Group of all Formal Finite Sums over Integer Ring
sage: FormalSums(ZZ)
Abelian Group of all Formal Finite Sums over Integer Ring
sage: FormalSums(GF(7))
Abelian Group of all Formal Finite Sums over Finite Field of size 7
sage: FormalSums(ZZ[sqrt(2)])
Abelian Group of all Formal Finite Sums over Order in Number Field in sqrt2 with defining polynomial x^2 - 2
sage: FormalSums(GF(9,'a'))
Abelian Group of all Formal Finite Sums over Finite Field in a of size 3^2

Element

alias of FormalSum

base_extend(R)

EXAMPLES:

sage: F7 = FormalSums(ZZ).base_extend(GF(7)); F7
Abelian Group of all Formal Finite Sums over Finite Field of size 7


The following tests against a bug that was fixed at trac ticket #18795:

sage: isinstance(F7, F7.category().parent_class)
True

sage.structure.formal_sum.formal_sums = Abelian Group of all Formal Finite Sums over Integer Ring