OpenMath Content Dictionary: field3

Canonical URL:

http://www.openmath.org/cd/field3.ocd

CD File:

field3.ocd

CD as XML Encoded OpenMath:

field3.omcd

Defines:

field_by_poly, fraction_field, free_field

Date:

2004-06-01

Version:

1 (Revision 1)

Review Date:

2006-06-01

Status:

experimental

A CD of functions for basic constructions in field theory.

Written by Arjeh M. Cohen 2004-02-25

free_field

Description:

This symbol represents a binary function. The first argument should be a natural number p which is zero or a prime number, the second argument a list or a set L. When evaluated on such arguments p and L, the function represents the field of rational functions in L over the rationals if p = 0 and over the field of integers mod p if p is a prime.

Example:

The rational function field Q(a,b) in the indeterminates a, b is

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="field3" name="free_field"/>
       <OMI>0</OMI>
       <OMA><OMS cd="list1" name="list"/>
            <OMV name="a"/>  <OMV name="b"/>
       </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="field3">free_field</csymbol>
  <cn>0</cn>
  <apply><csymbol cd="list1">list</csymbol><ci>a</ci><ci>b</ci></apply>
 </apply>
</math>

Prefix

free_field (0, list ( a, b) )

Popcorn

field3.free_field(0, [$a , $b])

Rendered Presentation MathML

$$\mathrm{free\_field}(0,(a,b))$$

Signatures:

sts

fraction_field

Description:

This is a unary function. Its argument should be a domain (as in CD ring4). It denotes the fraction field of the domain.

Example:

The rationals equals fraction_field(Integers)

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMS cd="fieldname1" name="Q"/>
  <OMA>
    <OMS cd="field3" name="fraction_field"/>
    <OMS cd="ringname1" name="Z"/>
  </OMA>
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <csymbol cd="fieldname1">Q</csymbol>
  <apply><csymbol cd="field3">fraction_field</csymbol><csymbol cd="ringname1">Z</csymbol></apply>
 </apply>
</math>

Prefix

eq (Q, fraction_field (Z) )

Popcorn

fieldname1.Q = field3.fraction_field(ringname1.Z)

Rendered Presentation MathML

$$Q=\mathrm{fraction\_field}\left(Z\right)$$

Signatures:

sts

field_by_poly

Description:

This symbol is a binary function whose first argument is a univariate polynomial ring R over a field, and whose second argument is an irreducible polynomial f in this polynomial ring R. So, when applied to R and f, the function has value the quotient ring R/(f).

Example:

The finite field GF(2)[X]/(X^2+X+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="field3" name="field_by_poly"/>
          <OMA id="pr"><OMS cd="polyd1" name="poly_ring_d_named"/>
               <OMA><OMS cd="setname2" name="GFp"/>
                    <OMI>2</OMI>
               </OMA>
               <OMV name="X"/>
          </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>1</OMI>
                    </OMA>
                    <OMA><OMS cd="polyd1" name="term"/>
                         <OMI>1</OMI><OMI>2</OMI>
                    </OMA>
               </OMA>
          </OMA>
     </OMA> 
</OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply>
  <apply><csymbol cd="field3">field_by_poly</csymbol>
   <apply id="pr"><csymbol cd="polyd1">poly_ring_d_named</csymbol>
    <apply><csymbol cd="setname2">GFp</csymbol><cn>2</cn></apply>
    <ci>X</ci>
   </apply>
   <apply><csymbol cd="polyd1">DMP</csymbol>
    <share href="#pr"/>
    <apply><csymbol cd="polyd1">SDMP</csymbol>
     <apply><csymbol cd="polyd1">term</csymbol><cn>1</cn><cn>0</cn></apply>
     <apply><csymbol cd="polyd1">term</csymbol><cn>1</cn><cn>1</cn></apply>
     <apply><csymbol cd="polyd1">term</csymbol><cn>1</cn><cn>2</cn></apply>
    </apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

field_by_poly (poly_ring_d_named (GFp (2) , X) , DMP (, SDMP (term (1, 0) , term (1, 1) , term (1, 2) ) ) ) ()

Popcorn

field3.field_by_poly(polyd1.poly_ring_d_named(setname2.GFp(2), $X):pr, polyd1.DMP(#pr, polyd1.SDMP(polyd1.term(1, 0), polyd1.term(1, 1), polyd1.term(1, 2))))()

Rendered Presentation MathML

$$\left(\mathrm{field\_by\_poly}(\mathrm{poly\_ring\_d\_named}({\mathbb{GF}}_2,X),\mathrm{DMP}(\mathrm{poly\_ring\_d\_named}({\mathbb{GF}}_2,X),\mathrm{SDMP}(\mathrm{term}(1,0),\mathrm{term}(1,1),\mathrm{term}(1,2))))\right)\left(\right)$$

or by

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
     <OMA><OMS cd="field3" name="field_by_poly"/>
          <OMA id="prn"><OMS cd="polyd1" name="poly_ring_d_named"/>
               <OMA><OMS cd="setname2" name="GFp"/>
                    <OMI>2</OMI>
               </OMA>
               <OMV name="X"/>
          </OMA>
          <OMA><OMS cd="ring1" name="expression"/>
               <OMR href="#prn"/>
               <OMA><OMS cd="arith1" name="plus"/>
                    <OMI>1</OMI>
                    <OMV name="X"/>
                    <OMA><OMS cd="arith1" name="power"/>
                         <OMV name="X"/><OMI>2</OMI>
                    </OMA>
               </OMA>
          </OMA>
     </OMA> 
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="field3">field_by_poly</csymbol>
  <apply id="prn"><csymbol cd="polyd1">poly_ring_d_named</csymbol>
   <apply><csymbol cd="setname2">GFp</csymbol><cn>2</cn></apply>
   <ci>X</ci>
  </apply>
  <apply><csymbol cd="ring1">expression</csymbol>
   <share href="#prn"/>
   <apply><csymbol cd="arith1">plus</csymbol>
    <cn>1</cn>
    <ci>X</ci>
    <apply><csymbol cd="arith1">power</csymbol><ci>X</ci><cn>2</cn></apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

field_by_poly (poly_ring_d_named (GFp (2) , X) , expression (, plus (1, X, power ( X, 2) ) ) )

Popcorn

field3.field_by_poly(polyd1.poly_ring_d_named(setname2.GFp(2), $X):prn, ring1.expression(#prn, 1 + $X + $X ^ 2))

Rendered Presentation MathML

$$\mathrm{field\_by\_poly}(\mathrm{poly\_ring\_d\_named}({\mathbb{GF}}_2,X),\mathrm{expression}(\mathrm{poly\_ring\_d\_named}({\mathbb{GF}}_2,X),1+X+X^2))$$

Signatures:

sts