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

Checks the ideals for various inconsistencies

<p><p>

<h3>
Syntax:
</h3>
<pre>errcount = IdealIntegrity(I);</pre>

<p>

<table>
<tr>
<td>ideal</td>
<td><pre>  I  </pre></td>
<td></td>
</tr>
<tr>
<td>small integer</td>
<td><pre>  errcount  </pre></td>
<td>the number of detected errors</td>
</tr>
</table>

<p>


<h3>
Description:
</h3>
The most important application of this function is this: Not
all matrices (square and with the correct dimension)
over the coefficient order of the ideal form an valid ideal.
If an ideal is created with a basis matrix it is not checked
if this matrix forms a correct ideal basis (because it is
much too expensive to check this for every ideal creation by
default.) Such an <em> wrong</em> ideal can cause all sorts of
follow-up errors which might be difficult to detect. If You
are not sure if the ideal is correct apply this function.

The ideal invariants which are stored in the ideal data
structure are checked for their validity as well, including
factorization, norm, minimum, principal generator, prime
element, primality.

If an inconsistency is detected a warning message is printed
to the screen but the break loop is not entered (except if it is
not possible to find a 2 element representation where the
break loop is entered). If the
return value is 0 the ideal is (very likely) correct.

<p><p>

<p>
<hr>
<p>
<h3>
Example:
</h3>
Correct ideal 
<pre>

kash> O := OrderMaximal(Order(Poly(Zx, [1, 6, 6, 6])));;
kash> I := Ideal(O, Mat(Z, [[2,0,0],[0,1,0],[0,0,1]]),1);
<
[2 0 0]
[0 1 0]
[0 0 1]
>

kash> IdealIntegrity(I);
> 0

</pre>

<p>


<hr>

<p>
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