OpenMath Content Dictionary: fieldname1

Canonical URL:

http://www.openmath.org/cd/fieldname1.ocd

CD Base:

http://www.openmath.org/cd

CD File:

fieldname1.ocd

CD as XML Encoded OpenMath:

fieldname1.omcd

Defines:

C, Q, R

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

C

Description:

This is a symbol representing the field of complex numbers.

Commented Mathematical property (CMP):

The field of complex numbers is (C, +,0,-,*,1,/), where +,-,*,/ are the standard arithmetic operations.

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="relation1" name="eq"/>
       <OMS cd="fieldname1" name="C"/>
       <OMA><OMS cd="field1" name="field"/>
            <OMS cd="setname1" name="C"/>
            <OMS cd="arith1" name="plus"/>
            <OMI>0</OMI>
            <OMS cd="arith1" name="minus"/>
            <OMS cd="arith1" name="times"/>
            <OMI>1</OMI>
            <OMS cd="arith1" name="divide"/>
       </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <csymbol cd="fieldname1">C</csymbol>
  <apply><csymbol cd="field1">field</csymbol>
   <csymbol cd="setname1">C</csymbol>
   <csymbol cd="arith1">plus</csymbol>
   <cn>0</cn>
   <csymbol cd="arith1">minus</csymbol>
   <csymbol cd="arith1">times</csymbol>
   <cn>1</cn>
   <csymbol cd="arith1">divide</csymbol>
  </apply>
 </apply>
</math>

Prefix

eq (C, field (C, plus, 0, minus, times, 1, divide) )

Popcorn

fieldname1.C = field1.field(setname1.C, arith1.plus, 0, arith1.minus, arith1.times, 1, arith1.divide)

Rendered Presentation MathML

$$C=\mathrm{field}(\mathbb{C},+,0,-,\times ,1,\frac{}{})$$

Signatures:

sts

R

Description:

This is a symbol representing the field of real numbers.

Commented Mathematical property (CMP):

The field of real numbers is (R, +,0,-,*,1,/), where +,-,*,/ are the standard arithmetic operations.

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="relation1" name="eq"/>
       <OMS cd="fieldname1" name="R"/>
       <OMA><OMS cd="field1" name="field"/>
                 <OMS cd="setname1" name="R"/>
                 <OMS cd="arith1" name="plus"/>
                 <OMI>0</OMI>
                 <OMS cd="arith1" name="minus"/>
                 <OMS cd="arith1" name="times"/>
                 <OMI>1</OMI>
                 <OMS cd="arith1" name="divide"/>
       </OMA>

  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <csymbol cd="fieldname1">R</csymbol>
  <apply><csymbol cd="field1">field</csymbol>
   <csymbol cd="setname1">R</csymbol>
   <csymbol cd="arith1">plus</csymbol>
   <cn>0</cn>
   <csymbol cd="arith1">minus</csymbol>
   <csymbol cd="arith1">times</csymbol>
   <cn>1</cn>
   <csymbol cd="arith1">divide</csymbol>
  </apply>
 </apply>
</math>

Prefix

eq (R, field (R, plus, 0, minus, times, 1, divide) )

Popcorn

fieldname1.R = field1.field(setname1.R, arith1.plus, 0, arith1.minus, arith1.times, 1, arith1.divide)

Rendered Presentation MathML

$$R=\mathrm{field}(\mathbb{R},+,0,-,\times ,1,\frac{}{})$$

Signatures:

sts

Q

Description:

This is a symbol representing the field of rational numbers.

Commented Mathematical property (CMP):

The field of rational numbers is (Q, +,0,-,*,1,/), where +,-,*,/ are the standard arithmetic operations.

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS cd="relation1" name="eq"/>
       <OMS cd="fieldname1" name="Q"/>
       <OMA><OMS cd="field1" name="field"/>
                 <OMS cd="setname1" name="Q"/>
                 <OMS cd="arith1" name="plus"/>
                 <OMI>0</OMI>
                 <OMS cd="arith1" name="minus"/>
                 <OMS cd="arith1" name="times"/>
                 <OMI>1</OMI>
                 <OMS cd="arith1" name="divide"/>
       </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="field1">field</csymbol>
   <csymbol cd="setname1">Q</csymbol>
   <csymbol cd="arith1">plus</csymbol>
   <cn>0</cn>
   <csymbol cd="arith1">minus</csymbol>
   <csymbol cd="arith1">times</csymbol>
   <cn>1</cn>
   <csymbol cd="arith1">divide</csymbol>
  </apply>
 </apply>
</math>

Prefix

eq (Q, field (Q, plus, 0, minus, times, 1, divide) )

Popcorn

fieldname1.Q = field1.field(setname1.Q, arith1.plus, 0, arith1.minus, arith1.times, 1, arith1.divide)

Rendered Presentation MathML

$$Q=\mathrm{field}(\mathbb{Q},+,0,-,\times ,1,\frac{}{})$$

Commented Mathematical property (CMP):

The carrier set of this field is the set of rational numbers.

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
  <OMA><OMS name="eq" cd="relation1"/>
       <OMA><OMS name="carrier" cd="ring1"/>
            <OMS name="Q" cd="fieldname1"/>
       </OMA>
       <OMS name="Q" cd="setname1"/>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="ring1">carrier</csymbol><csymbol cd="fieldname1">Q</csymbol></apply>
  <csymbol cd="setname1">Q</csymbol>
 </apply>
</math>

Prefix

eq (carrier (Q) , Q)

Popcorn

ring1.carrier(fieldname1.Q) = setname1.Q

Rendered Presentation MathML

$$\mathrm{carrier}\left(Q\right)=\mathbb{Q}$$

Example:

OpenMath XML (source)

  <OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
    <OMA><OMS cd="relation1" name="eq"/>
         <OMS cd="fieldname1" name="Q"/>
         <OMA><OMS cd="field1" name="field"/>
              <OMS cd="setname1" name="Q"/>
              <OMS cd="arith1" name="plus"/>
              <OMI>0</OMI>
              <OMS cd="arith1" name="minus"/>
              <OMS cd="arith1" name="times"/>
              <OMI>1</OMI>
              <OMBIND><OMS cd="fns1" name="lambda"/>
                   <OMBVAR>  <OMV name="x"/>  </OMBVAR>
                   <OMA><OMS cd="arith1" name="divide"/>
                        <OMI> 1 </OMI>  <OMV name="x"/>
                   </OMA>
              </OMBIND>
         </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="field1">field</csymbol>
   <csymbol cd="setname1">Q</csymbol>
   <csymbol cd="arith1">plus</csymbol>
   <cn>0</cn>
   <csymbol cd="arith1">minus</csymbol>
   <csymbol cd="arith1">times</csymbol>
   <cn>1</cn>
   <bind><csymbol cd="fns1">lambda</csymbol>
    <bvar><ci>x</ci></bvar>
    <apply><csymbol cd="arith1">divide</csymbol><cn>1</cn><ci>x</ci></apply>
   </bind>
  </apply>
 </apply>
</math>

Prefix

eq (Q, field (Q, plus, 0, minus, times, 1, lambda [ x ] . (divide ( 1 , x) ) ) )

Popcorn

fieldname1.Q = field1.field(setname1.Q, arith1.plus, 0, arith1.minus, arith1.times, 1, fns1.lambda[$x -> 1 / $x])

Rendered Presentation MathML

$$Q=\mathrm{field}(\mathbb{Q},+,0,-,\times ,1,\lambda \hspace{0.17em}x.\frac{1}{x})$$

Signatures:

sts