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A Tour of NTL: Examples: Extension Rings and Fields </title>
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A Tour of NTL: Examples: Extension Rings and Fields
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<p> <hr> <p>
NTL also supports extension rings and fields over finite fields,
and polynomial arithmetic over such extensions.
Here is a little program that illustrates this.
<p>
<pre>
#include <NTL/ZZ_pXFactoring.h>
#include <NTL/ZZ_pEX.h>
NTL_CLIENT
int main()
{
ZZ_p::init(to_ZZ(17)); // define GF(17)
ZZ_pX P;
BuildIrred(P, 10); // generate an irreducible polynomial P
// of degree 10 over GF(17)
ZZ_pE::init(P); // define GF(17^10)
ZZ_pEX f, g, h; // declare polynomials over GF(17^10)
random(f, 20); // f is a random, monic polynomial of degree 20
SetCoeff(f, 20);
random(h, 20); // h is a random polynomial of degree less than 20
g = MinPolyMod(h, f); // compute the minimum polynomial of h modulo f
if (g == 0) Error("oops (1)"); // check that g != 0
if (CompMod(g, h, f) != 0) // check that g(h) = 0 mod f
Error("oops (2)");
}
</pre>
<p>
This example illustrates building extension rings over <tt>ZZ_p</tt>.
One can also use <tt>zz_p</tt> and <tt>GF2</tt> as base classes;
the syntax is exactly the same.
<p>
See <a href="ZZ_pE.txt"><tt>ZZ_pE.txt</tt></a> for the basics of the extension
ring <tt>ZZ_pE</tt> over <tt>ZZ_p</tt>.
Also see <a href="ZZ_pEX.txt"><tt>ZZ_pEX.txt</tt></a> for polynomial
arithmetic over <tt>ZZ_pE</tt>, and
<a href="ZZ_pEXFactoring.txt"><tt>ZZ_pEXFactoring.txt</tt></a> for factoring
routines over <tt>ZZ_pE</tt>.
See <a href="vec_ZZ_pE.txt"><tt>vec_ZZ_pE.txt</tt></a> for vectors over <tt>ZZ_pE</tt>,
and <a href="mat_ZZ_pE.txt"><tt>mat_ZZ_pE.txt</tt></a> for matrices over <tt>ZZ_pE</tt>.
<p>
See <a href="lzz_pE.txt"><tt>lzz_pE.txt</tt></a> for the basics of the extension
ring <tt>zz_pE</tt> over <tt>zz_p</tt>.
Also see <a href="lzz_pEX.txt"><tt>lzz_pEX.txt</tt></a> for polynomial
arithmetic over <tt>zz_pE</tt>, and
<a href="lzz_pEXFactoring.txt"><tt>lzz_pEXFactoring.txt</tt></a> for factoring
routines over <tt>zz_pE</tt>.
See <a href="vec_lzz_pE.txt"><tt>vec_lzz_pE.txt</tt></a> for vectors over <tt>zz_pE</tt>,
and <a href="mat_lzz_pE.txt"><tt>mat_lzz_pE.txt</tt></a> for matrices over <tt>zz_pE</tt>.
<p>
See <a href="GF2E.txt"><tt>GF2E.txt</tt></a> for the basics of the extension
ring <tt>GF2E</tt> over <tt>GF2</tt>.
Also see <a href="GF2EX.txt"><tt>GF2EX.txt</tt></a> for polynomial
arithmetic over <tt>GF2E</tt>, and
<a href="GF2EXFactoring.txt"><tt>GF2EXFactoring.txt</tt></a> for factoring
routines over <tt>GF2E</tt>.
See <a href="vec_GF2E.txt"><tt>vec_GF2E.txt</tt></a> for vectors over <tt>GF2E</tt>,
and <a href="mat_GF2E.txt"><tt>mat_GF2E.txt</tt></a> for matrices over <tt>GF2E</tt>.
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