OpenMath Content Dictionary: fieldname1

Canonical URL:

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

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):

xml prefix mathml

<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>

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

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

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):

xml prefix mathml

<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>

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

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

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):

xml prefix mathml

<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>

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

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

Commented Mathematical property (CMP):

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

Formal Mathematical property (FMP):

xml prefix mathml

<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>

eq (carrier (Q) , Q)

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

Example:

xml prefix mathml

<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>

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

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

Signatures:

sts