Minpoly - Maple Help

evala/Minpoly

minimal polynomial of an algebraic number (or function)

 Calling Sequence evala(Minpoly(a, x, K))

Parameters

 a - algebraic number or function x - name K - (optional) set of algebraic numbers or functions defining an extension field

Description

 • The Minpoly function is a placeholder for representing the minimal polynomial of an algebraic number (or function) $a$. It is used in conjunction with evala.
 • The call evala(Minpoly(a, x)) computes the monic minimal polynomial of $a$ in the variable $x$ over the field of rational numbers (or multivariate rational functions). The resulting polynomial will not contain any algebraic numbers or functions.
 • The call evala(Minpoly(a, x, K)) computes the monic minimal polynomial of $a$ in the variable $x$ over the field $F$ resulting from an extension of the rational numbers (or multivariate rational functions) by the algebraic numbers (or functions) in $K$. The resulting polynomial will only contain algebraic numbers or functions from $F$ in its coefficients.
 • The variable $x$ cannot occur in either $a$ or $K$; otherwise, an error will be raised.
 • The algebraic numbers and functions in both $a$ and $K$ can be given either in radical or RootOf notation. A mixture or radicals and RootOfs is not supported. The coefficients of the resulting polynomial will be returned in the same form as $K$ (if specified).
 • If the algebraic numbers and functions in $K$ do not form a syntactical subset of the algebraic numbers and functions occurring in $a$, evala/Algfield will be used to rewrite $a$ as an element of an appropriate extension field of $F$. This may not always succeed, and as a result, the polynomial returned may not be of minimal degree in that case.

Examples

 > $\mathrm{alias}\left(\mathrm{sqrt2}=\mathrm{RootOf}\left({x}^{2}-2,\mathrm{index}=1\right)\right):$
 > $\mathrm{alias}\left(i=\mathrm{RootOf}\left({x}^{2}+1,\mathrm{index}=1\right)\right):$
 > $\mathrm{alias}\left(\mathrm{sqrty}=\mathrm{RootOf}\left({x}^{2}-y,x\right)\right):$
 > $\mathrm{evala}\left(\mathrm{Minpoly}\left(1+\mathrm{sqrt2},x\right)\right)$
 ${{x}}^{{2}}{-}{2}{}{x}{-}{1}$ (1)
 > $\mathrm{evala}\left(\mathrm{Minpoly}\left(\mathrm{sqrt2}+\mathrm{sqrty},x\right)\right)$
 ${{x}}^{{4}}{-}{2}{}\left({y}{+}{2}\right){}{{x}}^{{2}}{+}{{y}}^{{2}}{-}{4}{}{y}{+}{4}$ (2)
 > $\mathrm{evala}\left(\mathrm{Minpoly}\left(\mathrm{sqrt2}\mathrm{sqrty},x\right)\right)$
 ${{x}}^{{2}}{-}{2}{}{y}$ (3)

Specifying an extension field.

 > $\mathrm{alias}\left(\mathrm{\alpha }=\mathrm{RootOf}\left({x}^{4}+1,\mathrm{index}=1\right)\right):$
 > $\mathrm{evala}\left(\mathrm{Minpoly}\left(\mathrm{\alpha },x,i\right)\right)$
 ${{x}}^{{2}}{-}{i}$ (4)
 > $\mathrm{evala}\left(\mathrm{Minpoly}\left(\mathrm{\alpha },x,\mathrm{sqrt2}\right)\right)$
 ${-}{x}{}{\mathrm{sqrt2}}{+}{{x}}^{{2}}{+}{1}$ (5)

 > $\mathrm{evala}\left(\mathrm{Minpoly}\left({\left(-1\right)}^{\frac{1}{4}},x,I\right)\right)$
 ${{x}}^{{2}}{-}{I}$ (6)
 > $f≔\mathrm{evala}\left(\mathrm{Minpoly}\left({\left(-1\right)}^{\frac{1}{4}},x,\mathrm{sqrt}\left(2\right)\right)\right)$
 ${f}{≔}{-}{x}{}\sqrt{{2}}{+}{{x}}^{{2}}{+}{1}$ (7)
 > $\mathrm{eval}\left(f,x={\left(-1\right)}^{\frac{1}{4}}\right)$
 ${-}{\left({-1}\right)}^{{1}}{{4}}}{}\sqrt{{2}}{+}{1}{+}{I}$ (8)
 > $\mathrm{radnormal}\left(\right)$
 ${0}$ (9)

If the algebraic numbers and functions are not independent, i.e., they do not form a field, or if the algebraic numbers and functions in $K$ do not occur in $a$, the resulting polynomial may not be of minimal degree. In the following example, Maple is unable to tell whether ${\mathrm{\beta }}^{2}$ equals sqrt2 or -sqrt2.

 > $\mathrm{alias}\left(\mathrm{\beta }=\mathrm{RootOf}\left({x}^{4}-2\right)\right):$
 > $\mathrm{evala}\left(\mathrm{Minpoly}\left(\mathrm{\beta },x,\mathrm{sqrt2}\right)\right)$
 ${{x}}^{{4}}{-}{2}$ (10)

In this example, using an indexed RootOf will help.

 > $\mathrm{evala}\left(\mathrm{Minpoly}\left(\mathrm{RootOf}\left({x}^{4}-2,\mathrm{index}=1\right),x,\mathrm{sqrt2}\right)\right)$
 ${{x}}^{{2}}{-}{\mathrm{sqrt2}}$ (11)
 > $\mathrm{evala}\left(\mathrm{Minpoly}\left(\mathrm{RootOf}\left({x}^{4}-2,\mathrm{index}=2\right),x,\mathrm{sqrt2}\right)\right)$
 ${{x}}^{{2}}{+}{\mathrm{sqrt2}}$ (12)

Compatibility

 • The evala/Minpoly command was introduced in Maple 2020.