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<p>&nbsp;<p>
<hr noshade size="5">
<h2><a name="AlffResFieldEltLift"></a>
AlffResFieldEltLift
</h2>

Lifts a residue field element.

<p><p>

<h3>
Syntax:
</h3>
<pre>a := AlffResFieldEltLift(b, P);</pre>

<p>

<table>
<tr>
<td>alff element</td>
<td><pre>  a  </pre></td>
<td>in o_P</td>
</tr>
<tr>
<td>field element</td>
<td><pre>  b  </pre></td>
<td>value of a at P to be lifted</td>
</tr>
<tr>
<td>alff place </td>
<td><pre>  P  </pre></td>
<td></td>
</tr>
</table>

<p>


See also:&nbsp;&nbsp;<a href="kashref.html#AlffResFieldEltLift">AlffResFieldEltLift</a>, <a href="kashref.html#AlffELtToResField">AlffELtToResField</a>, <a href="kashref.html#AlffEltValuation">AlffEltValuation</a>

<p>

<h3>
Description:
</h3>
Let F/k be an algebraic function field and P be a place
of F/k. If o_P is the ring of algebraic functions defined
at P then k(P) := o_P / P is the residue class field
of P. For b \in k(P) this function returns a \in o_P
such that a(P) = b. Constants are lifted to constants.

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>

<pre>

kash> AlffInit(FF(5,3));
"Defining global variables: k, w, kT, kTf, kTy, T, y, AlffGlobals"
kash> F := Alff(y^3+T^3*y+T);
Algebraic function field defined by
.1^3 + .1*.2^3 + .2
over
Univariate rational function field over GF(5^3)
Variables: T

kash> P := AlffPlaceSplit(F, T+1)[1];
Alff place < [ T + 1, 0, 0 ], [ 3, 1, 0 ] >
kash> AlffPlaceResField(P);
Finite field of size 5^3
kash> of := AlffOrderMaxFinite(F);
Finite maximal order of 
Algebraic function field defined by
.1^3 + .1*.2^3 + .2
over
Univariate rational function field over GF(5^3)
Variables: T

kash> a := AlffEltToResField(AlffElt(of,T),P);
4
kash> AlffResFieldEltLift(a,P);
[ 4, 0, 0 ]
kash> p := AlffPlaceSplit(F, 1/T)[1];
Alff place < [ 1/T, 0, 0 ], [ w^48/T, (w^102*T + w^5)/T, (w^18*T + w^82)/T ] >
kash> AlffPlaceResField(p);
Finite field of size 5^3
kash> oi := AlffOrderMaxInfty(F);
Infinite maximal order of 
Algebraic function field defined by
.1^3 + .1*.2^3 + .2
over
Univariate rational function field over GF(5^3)
Variables: T
given by transformation matrix
[1/T   0   0]
[  0 1/T   0]
[  0   0   1]
with denominator 1/T
kash> b := AlffEltToResField(AlffElt(oi,1/T),p);
0
kash> AlffResFieldEltLift(b,p);
[ 0, 0, 0 ]

kash> AlffInit(Q);
"Defining global variables: k, w, kT, kTf, kTy, T, y, AlffGlobals"
kash> F := Alff(y^3+T^3*y+T);
Algebraic function field defined by
.1^3 + .1*.2^3 + .2
over
Univariate rational function field over Rational Field
Variables: T

kash> P := AlffPlaceSplit(F, T+1)[1];
Alff place < [ T + 1, 0, 0 ] >
kash> AlffPlaceResField(P);
Generating polynomial: x^3 - x - 1

kash> of := AlffOrderMaxFinite(F);
Finite maximal order of 
Algebraic function field defined by
.1^3 + .1*.2^3 + .2
over
Univariate rational function field over Rational Field
Variables: T

kash> a := AlffEltToResField(AlffElt(of,2),P);
2
kash> AlffResFieldEltLift(a,P);
[ 2, 0, 0 ]
kash> p := AlffPlaceSplit(F, 1/T)[1];
Alff place < [ 1/T, 0, 0 ], [ -13/T, (11/3*T - 35)/T, (-3/2*T + 32/3)/T ] >
kash> AlffPlaceResField(p);
Rational Field

kash> oi := AlffOrderMaxInfty(F);
Infinite maximal order of 
Algebraic function field defined by
.1^3 + .1*.2^3 + .2
over
Univariate rational function field over Rational Field
Variables: T
given by transformation matrix
[1/T   0   0]
[  0 1/T   0]
[  0   0   1]
with denominator 1/T
kash> b := AlffEltToResField(AlffElt(oi,1/T),p);
0
kash> AlffResFieldEltLift(b,p);
> [ 0, 0, 0 ]

</pre>

<p>


<hr>

<p>
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