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<a name="Introduction-to-algebraic-extensions"></a>
<div class="header">
<p>
Next: <a href="maxima_48.html#Functions-and-Variables-for-algebraic-extensions" accesskey="n" rel="next">Functions and Variables for algebraic extensions</a>, Previous: <a href="maxima_46.html#Functions-and-Variables-for-Polynomials" accesskey="p" rel="previous">Functions and Variables for Polynomials</a>, Up: <a href="maxima_44.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_264.html#g_t_0423_043a_0430_0437_0430_0442_0435_043b_044c-_0444_0443_043d_043a_0446_0438_0439-_0438-_043f_0435_0440_0435_043c_0435_043d_043d_044b_0445" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Introduction-to-algebraic-extensions-1"></a>
<h3 class="section">11.3 Introduction to algebraic extensions</h3>
<p>We assume here that the fields are of characteristic 0 so that
irreductible polynomials have simple roots (are separable, thus square
free). The base fields <em>K</em> of interest are the field <em>Q</em> of rational
numbers, for algebraic numbers, and the fields of rational functions on
the real numbers <em>R</em> or the complex numbers <em>C</em>, that is <em>R(t)</em> or <em>C(t)</em>, when
considering algebraic functions. An extension of degree <em>n</em> is defined by
an irreducible degree <em>n</em> polynomial <em>p(x)</em> with coefficients in the base
field, and consists of the quotient of the ring <em>K[x]</em> of polynomials by
the multiples of <em>p(x)</em>. So if <em>p(x) = x^n + p_0 x^{n - 1} + ... + p_n</em>, each time one
encounters <em>x^n</em> one substitutes <em>-(p_0 x^{n - 1} + ... + p_n)</em>. This is a field
because of Bezout’s identity, and a vector space of dimension <em>n</em> over <em>K</em>
spanned by <em>1, x, ..., x^{n - 1}</em>. When <em>K = C(t)</em>, this field can be identified
with the field of algebraic functions on the algebraic curve of equation
<em>p(x, t) = 0</em>.
</p>
<p>In Maxima the process of taking rationals modulo <em>p</em> is obtained by the
function <code>tellrat</code> when <code>algebraic</code> is true. The best way to ensure,
in particular when considering the case where <em>p</em> depends on other
variables that this simplification property is attached to <em>x</em> is to write
(note the polynomial must be monic):
<code>tellrat(x^n = -(p_0*x^(n - 1) + ... + p_n))</code> where the <em>p_i</em> may depend on
other variables. When one wants to remove this tellrat property one then
has to write <code>untellrat(x)</code>.
</p>
<p>In the field <em>K[x]</em> one may do all sorts of algebraic computations, taking
quotients, GCD of two elements, etc. by the same algorithms as in the
usual case. In particular one can do factorization of polynomials on an
extension, using the function <code>algfac</code> below. Moreover
multiplication by an element <em>f</em> is a linear operation of the vector space
<em>K[x]</em> over <em>K</em> and as such has a trace and a determinant. These are called
<code>algtrace</code> and <code>algnorm</code> below. One can see that the trace of
an element <em>f(x)</em> in <em>K[x]</em> is the sum of the values <em>f(a)</em> when <em>a</em> runs over
roots of <em>p</em> and the norm is the product of the <em>f(a)</em>. Both are symmetric
in the roots of <em>p</em> and thus belong to <em>K</em>.
</p>
<p>The field <em>K[x]</em> is also called the field obtained by adjoining a root <em>a</em>
of <em>p(x)</em> to <em>K</em>. One can similarly adjoin a second root <em>b</em> of another
polynomial obtaining a new extension <em>K[a,b]</em>. In fact there is a “prime
element” <em>c</em> in <em>K[a, b]</em> such that <em>K[a, b] = K[c]</em>. This is obtained by
function <code>primeelmt</code> below. Recursively one can thus adjoin any
number of elements. In particular adjoining all the roots of <em>p(x)</em> to <em>K</em>
one gets the splitting field of <em>p</em>, which is the smallest extension in
which <em>p</em> completely splits in linear functions. The function
<code>splitfield</code> constructs a primitive element of the splitting field,
which in general is of very high degree.
</p>
<p>The relevant concepts are explained in a concise and self-contained way in the
small books edited by Dover:
“Algebraic theory of numbers,” by Pierre Samuel,
“Algebraic curves,” by Robert Walker,
and the methods presented here are described in the article
“Algebraic factoring and rational function integration” by B. Trager,
<em>Proceedings of the 1976 AMS Symposium on Symbolic and Algebraic Computation</em>.
</p>
<hr>
<div class="header">
<p>
Next: <a href="maxima_48.html#Functions-and-Variables-for-algebraic-extensions" accesskey="n" rel="next">Functions and Variables for algebraic extensions</a>, Previous: <a href="maxima_46.html#Functions-and-Variables-for-Polynomials" accesskey="p" rel="previous">Functions and Variables for Polynomials</a>, Up: <a href="maxima_44.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_264.html#g_t_0423_043a_0430_0437_0430_0442_0435_043b_044c-_0444_0443_043d_043a_0446_0438_0439-_0438-_043f_0435_0440_0435_043c_0435_043d_043d_044b_0445" title="Index" rel="index">Index</a>]</p>
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