OpenMath Content Dictionary: field4
Canonical URL:
http://www.openmath.org/cd/field4.ocd
CD Base:
http://www.openmath.org/cd
CD File:
field4.ocd
CD as XML Encoded OpenMath:
field4.omcd
Defines:
automorphism_group , field_by_poly_map , field_by_poly_vector , homomorphism_by_generators
Date:
2004-06-01
Version:
1
(Revision 1)
Review Date:
2006-06-01
Status:
experimental
A CD of
functions for morphisms of fields.
Written by Arjeh M. Cohen 2004-07-07
Description:
This is a function with a single argument which must be a field.
It refers to the automorphism group of its argument.
Signatures:
sts
Description:
This is a function with three arguments the first two of which must be fields
F and K.
The third argument should be a set or a list L of ordered pairs (lists of length 2). Each
pair [x,y] from L consists of an element x from F and an element y from K.
when applied to F, K, and L, the symbol represents the homomorphism from F
to K that maps the first entry x of each pair [x,y] to the second entry y of the same pair.
Signatures:
sts
Description:
Same as quotient_by_poly_map in CD ring5, except that R and the quotient ring R[X]/(f) are
fields (so f is irreducible in R[X]).
Example:
An element aX + b of the finite field GF(3)[X]/(X^2+1) is represented by
OpenMath XML (source)
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA><OMA><OMS cd="field4" name="field_by_poly_map"/>
<OMA id="pr"><OMS cd="polyd1" name="poly_ring_d"/>
<OMA><OMS cd="setname2" name="GFp"/>
<OMI>3</OMI>
</OMA>
<OMI>1</OMI>
</OMA>
<OMA><OMS cd="polyd1" name="DMP"/>
<OMR href="#pr"/>
<OMA><OMS cd="polyd1" name="SDMP"/>
<OMA><OMS cd="polyd1" name="term"/>
<OMI>1</OMI><OMI>0</OMI>
</OMA>
<OMA><OMS cd="polyd1" name="term"/>
<OMI>1</OMI><OMI>2</OMI>
</OMA>
</OMA>
</OMA>
</OMA>
<OMA><OMS cd="polyd1" name="DMP"/>
<OMR href="#pr"/>
<OMA><OMS cd="polyd1" name="SDMP"/>
<OMA><OMS cd="polyd1" name="term"/>
<OMV name="b"/><OMI>0</OMI>
</OMA>
</OMA>
<OMA><OMS cd="polyd1" name="term"/>
<OMV name="a"/><OMI>1</OMI>
</OMA>
</OMA>
</OMA>
</OMOBJ>
Strict Content MathML
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<apply><csymbol cd="field4">field_by_poly_map</csymbol>
<apply id="pr"><csymbol cd="polyd1">poly_ring_d</csymbol>
<apply><csymbol cd="setname2">GFp</csymbol><cn type="integer">3</cn></apply>
<cn type="integer">1</cn>
</apply>
<apply><csymbol cd="polyd1">DMP</csymbol>
<share src="#pr"/>
<apply><csymbol cd="polyd1">SDMP</csymbol>
<apply><csymbol cd="polyd1">term</csymbol>
<cn type="integer">1</cn>
<cn type="integer">0</cn>
</apply>
<apply><csymbol cd="polyd1">term</csymbol>
<cn type="integer">1</cn>
<cn type="integer">2</cn>
</apply>
</apply>
</apply>
</apply>
<apply><csymbol cd="polyd1">DMP</csymbol>
<share src="#pr"/>
<apply><csymbol cd="polyd1">SDMP</csymbol>
<apply><csymbol cd="polyd1">term</csymbol><ci>b</ci><cn type="integer">0</cn></apply>
</apply>
<apply><csymbol cd="polyd1">term</csymbol><ci>a</ci><cn type="integer">1</cn></apply>
</apply>
</apply>
</math>
Prefix
field_by_poly_map
(
poly_ring_d
(
GFp
(3)
, 1)
,
DMP
(,
SDMP
(
term
(1, 0)
,
term
(1, 2)
)
)
)
(
DMP
(,
SDMP
(
term
(
b , 0)
)
,
term
(
a , 1)
)
)
Popcorn
field4.field_by_poly_map(polyd1.poly_ring_d(setname2.GFp(3), 1):pr, polyd1.DMP(#pr, polyd1.SDMP(polyd1.term(1, 0), polyd1.term(1, 2))))(polyd1.DMP(#pr, polyd1.SDMP(polyd1.term($b, 0)), polyd1.term($a, 1)))
Rendered Presentation MathML
(
field_by_poly_map
(
poly_ring_d
(
GF
3
,
1
)
,
DMP
(
poly_ring_d
(
GF
3
,
1
)
,
SDMP
(
term
(
1
,
0
)
,
term
(
1
,
2
)
)
)
)
)
(
DMP
(
poly_ring_d
(
GF
3
,
1
)
,
SDMP
(
term
(
b
,
0
)
)
,
term
(
a
,
1
)
)
)
Signatures:
sts
Description:
This symbol is a binary function. Its first argument should be
a field_by_poly(R,f). Its second argument should be a
list L of elements of F, the coefficient field of the univariate polynomial
ring R = F[X].
The length of the list L should be equal to the degree d of f.
When applied to R and
L, it represents the element L[0] + L[1] x + L[2] x^2 + ... + L[d-1] ^(d-1) of
R/(f),
where x stands for the image of x under the natural quotient map R -> R/(f).
If the first argument is a field_by_conway(p,n), defined in the CD finfield1, then
the same interpretation holds, where R and f are respectively poly_ring_d(GFp(p),1) and conway_polynomial(p,n).
Commented Mathematical property (CMP):
later
Example:
The element x+1 of the Conway field of order 4:
OpenMath XML (source)
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
<OMA><OMS cd="field4" name="field_by_poly_vector"/>
<OMA><OMS cd="finfield1" name="field_by_conway"/>
<OMA><OMS cd="setname2" name="GFpn"/>
<OMI>2</OMI><OMI>2</OMI>
</OMA>
</OMA>
<OMA><OMS cd="list1" name="list"/>
<OMI>1</OMI>
<OMI>1</OMI>
</OMA>
</OMA>
</OMOBJ>
Strict Content MathML
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply><csymbol cd="field4">field_by_poly_vector</csymbol>
<apply><csymbol cd="finfield1">field_by_conway</csymbol>
<apply><csymbol cd="setname2">GFpn</csymbol>
<cn type="integer">2</cn>
<cn type="integer">2</cn>
</apply>
</apply>
<apply><csymbol cd="list1">list</csymbol>
<cn type="integer">1</cn>
<cn type="integer">1</cn>
</apply>
</apply>
</math>
Prefix
Popcorn
field4.field_by_poly_vector(finfield1.field_by_conway(setname2.GFpn(2, 2)), [1 , 1])
Rendered Presentation MathML
field_by_poly_vector
(
field_by_conway
(
GF
2
2
)
,
(
1
,
1
)
)
Signatures:
sts
