OpenMath Content Dictionary: field4

Canonical URL:

http://www.openmath.org/cd/field4.ocd

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

automorphism_group

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

homomorphism_by_generators

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

field_by_poly_map

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

xml prefix mathml

<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>

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) ) )

$$\left(\mathrm{field\_by\_poly\_map}(\mathrm{poly\_ring\_d}({\mathbb{GF}}_3,1),\mathrm{DMP}(,\mathrm{SDMP}(\mathrm{term}(1,0),\mathrm{term}(1,2))))\right)\left(\mathrm{DMP}(,\mathrm{SDMP}\left(\mathrm{term}(b,0)\right),\mathrm{term}(a,1))\right)$$

Signatures:

sts

field_by_poly_vector

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:

xml prefix mathml

<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>

field_by_poly_vector (field_by_conway (GFpn (2, 2) ) , list (1, 1) )

$$\mathrm{field\_by\_poly\_vector}(\mathrm{field\_by\_conway}\left({\mathbb{GF}}_{2^2}\right),(1,1))$$

Signatures:

sts