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<a name="Functions-and-Variables-for-algebraic-extensions"></a>
<div class="header">
<p>
Previous: <a href="maxima_81.html#Introduction-to-algebraic-extensions" accesskey="p" rel="previous">Introduction to algebraic extensions</a>, Up: <a href="maxima_78.html#Polynomials" accesskey="u" rel="up">Polynomials</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Functions-and-Variables-for-algebraic-extensions-1"></a>
<h3 class="section">14.4 Functions and Variables for algebraic extensions</h3>
<a name="algfac"></a><dl>
<dt><a name="index-algfac"></a>Function: <strong>algfac</strong> <em>(<var>f</var>, <var>p</var>)</em></dt>
<dd>
<p>Returns the factorization of <var>f</var> in the field <em>K[a]</em>. Does the same
as <code>factor(<var>f</var>, <var>p</var>)</code> which in fact calls <code>algfac</code>. One can also
specify the variable <var>a</var> as in <code>algfac(<var>f</var>, <var>p</var>, <var>a</var>)</code>.
</p>
<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) algfac(x^4 + 1, a^2 - 2);
2 2
(%o1) (x - a x + 1) (x + a x + 1)
(%i2) algfac(x^4 - t*x^2 + 1, a^2 - t - 2, a);
2 2
(%o2) (x - a x + 1) (x + a x + 1)
</pre></div>
<p>In the second example note that <em>a = sqrt(2 + t)</em>.
</p></dd></dl>
<a name="algnorm"></a><dl>
<dt><a name="index-algnorm"></a>Function: <strong>algnorm</strong> <em>(<var>f</var>, <var>p</var>, <var>a</var>)</em></dt>
<dd>
<p>Returns the norm of the polynomial <em>f(a)</em> in the extension
obtained by a root <var>a</var> of polynomial <var>p</var>. The coefficients of
<var>f</var> may depend on other variables.
</p>
<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) algnorm(x*a^2 + y*a + z,a^2 - 2, a);
2 2 2
(%o1)/R/ z + 4 x z - 2 y + 4 x
</pre></div>
<p>The norm is also the resultant of polynomials <var>f</var> and <var>p</var>, and the product
of the differences of the roots of <var>f</var> and <var>p</var>.
</p></dd></dl>
<a name="algtrace"></a><dl>
<dt><a name="index-algtrace"></a>Function: <strong>algtrace</strong> <em>(<var>f</var>, <var>p</var>, <var>a</var>)</em></dt>
<dd>
<p>Returns the trace of the polynomial <em>f(a)</em> in the extension
obtained by a root <var>a</var> of polynomial <var>p</var>. The coefficients of
<var>f</var> may depend on other variables which remain “inert”.
</p>
<p>Example:
</p>
<div class="example">
<pre class="example">(%i1) algtrace(x*a^5 + y*a^3 + z + 1, a^2 + a + 1, a);
(%o1)/R/ 2 z + 2 y - x + 2
</pre></div>
</dd></dl>
<a name="bdiscr"></a><dl>
<dt><a name="index-bdiscr"></a>Function: <strong>bdiscr</strong> <em>(<var>args</var>)</em></dt>
<dd>
<p>Computes the discriminant of a basis <em>x_i</em> in <em>K[a]</em> as
the determinant of the matrix of elements <em>trace(x_i*x_j)</em>.
The args are the elements of the basis followed by the minimal
polynomial.
</p>
<p>Example:
</p>
<div class="example">
<pre class="example">(%i1) bdiscr(1, x, x^2, x^3 - 2);
(%o1)/R/ - 108
(%i2) poly_discriminant(x^3 - 2, x);
(%o2) - 108
</pre></div>
<p>A standard base in an extension of degree n is <em>1, x, ..., x^{n - 1}</em>.
In this case it is known that the discriminant of this base is the discriminant
of the minimal polynomial. This is checked in (%o2) above.
</p>
</dd></dl>
<a name="primelmt"></a><dl>
<dt><a name="index-primelmt"></a>Function: <strong>primelmt</strong> <em>(<var>f_b</var>, <var>p_a</var>, <var>c</var>)</em></dt>
<dd>
<p>Computes a prime element for the extension of <em>K[a]</em> by a root
<var>b</var> of a polynomial <em>f_b(b)</em> whose coefficients may depend on
<var>a</var>. One assumes that <var>f_b</var> is square free. The function returns
an irreducible polynomial, a root of which generates <em>K[a, b]</em>, and
the expression of this primitive element in terms of <var>a</var> and
<var>b</var>.
</p>
<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) primelmt(b^2 - a*b - 1, a^2 - 2, c);
4 2
(%o1) [c - 12 c + 9, b + a]
(%i2) solve(b^2 - sqrt(2)*b - 1)[1];
sqrt(6) - sqrt(2)
(%o2) b = - -----------------
2
(%i3) primelmt(b^2 - 3, a^2 - 2, c);
4 2
(%o3) [c - 10 c + 1, b + a]
(%i4) factor(c^4 - 12*c^2 + 9, a^4 - 10*a^2 + 1);
3 2 3 2
(%o4) ((4 c - 3 a - a + 27 a + 5) (4 c - 3 a + a + 27 a - 5)
3 2 3 2
(4 c + 3 a - a - 27 a + 5) (4 c + 3 a + a - 27 a - 5))/256
(%i5) primelmt(b^3 - 3, a^2 - 2, c);
6 4 3 2
(%o5) [c - 6 c - 6 c + 12 c - 36 c + 1, b + a]
(%i6) factor(b^3 - 3, %[1]);
5 4 3 2
(%o6) ((48 c + 27 c - 320 c - 468 c + 124 c + 755 b - 1092)
5 5 4 4 3 3 2 2
((- 48 b c ) - 54 c - 27 b c + 64 c + 320 b c + 360 c + 468 b c + 149 c
2
- 124 b c - 1272 c + 755 b + 1092 b + 1606))/570025
</pre></div>
<p>In (%o1), <var>f_b</var> depends on <code>a</code>. Using <code>solve</code>, the solution depends on sqrt(2) and sqrt(3).
In (%o3), <em>K[sqrt(2), sqrt(3)]</em> is computed, and we see that the the primitive polynomial
in (%o1) factorizes completely here. In (%i5), we compute <em>K[sqrt(2), 3^{1/3}]</em>, and we see
that <code>b^3 - 3</code> gets one factor in this extension. If we assume this extension is real,
the two other factors are complex.
</p>
</dd></dl>
<a name="splitfield"></a><dl>
<dt><a name="index-splitfield"></a>Function: <strong>splitfield</strong> <em>(<var>p</var>, <var>x</var>)</em></dt>
<dd>
<p>Computes the splitting field of the polynomial <em>p(x)</em>.
In the generic case it is of degree <em>n!</em> in terms of the degree <em>n</em>
of <var>p</var>, but may be of lower order if the Galois group of <var>p</var>
is a strict subgroup of the group of permutations of <em>n</em>
elements. The function returns a primitive polynomial for this extension
and the expressions of the roots of <var>p</var> as polynomials of a root
of this primitive polynomial. The polynomial <var>f</var> may be
irreducible or factorizable.
</p>
<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) splitfield(x^3 + x + 1, x);
4 2
6 4 2 alg1 + 5 alg1 - 9 alg1 + 4
(%o1)/R/ [alg1 + 6 alg1 + 9 alg1 + 31, ----------------------------,
18
4 2 4 2
alg1 + 5 alg1 + 4 alg1 + 5 alg1 + 9 alg1 + 4
- -------------------, ----------------------------]
9 18
(%i2) splitfield(x^4 + 10*x^2 - 96*x - 71, x)[1];
8 6 5 4 3
(%o2)/R/ alg2 + 148 alg2 - 576 alg2 + 9814 alg2 - 42624 alg2
2
+ 502260 alg2 + 1109952 alg2 + 18860337
</pre></div>
<p>In the first case we have the primitive polynomial of degree 6 and the 3 roots
of the third degree equations in terms of a variable <code>alg1</code> produced by
the system. In the second case the primitive polynomial is of degree 8
instead of 24, because the Galois group of the equation is reduced to D8
since there are relations between the roots.
</p>
</dd></dl>
<hr>
<div class="header">
<p>
Previous: <a href="maxima_81.html#Introduction-to-algebraic-extensions" accesskey="p" rel="previous">Introduction to algebraic extensions</a>, Up: <a href="maxima_78.html#Polynomials" accesskey="u" rel="up">Polynomials</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
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