master1729 <tommy1...@gmail.com> writes: > > what do you know about cos(pi/19) ?
> > nice expressions or properties ?
> algebraic order is 9.
Indeed, its minimal polynomial over the rationals is -1+10*x+40*x^2-160*x^3-240*x^4+672*x^5+448*x^6-1024*x^7-256*x^8+512*x^9 which might look a bit nicer as -1+5*z+10*z^2-20*z^3-15*z^4+21*z^5+7*z^6-8*z^7-z^8+z^9 where z = 2*x. Thus 2*cos(pi/19) is an algebraic integer (which is also obvious from the fact that it is exp(i pi/19) + exp(-i pi/19), the sum of two roots of unity).
Its conjugates are cos(3 pi/19), cos(5 pi/19), ..., cos(17 pi/19). -- Robert Israel isr...@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
<isr...@math.MyUniversitysInitials.ca> wrote: > master1729 <tommy1...@gmail.com> writes: > > > what do you know about cos(pi/19) ?
> > > nice expressions or properties ?
> > algebraic order is 9.
> Indeed, its minimal polynomial over the rationals is > -1+10*x+40*x^2-160*x^3-240*x^4+672*x^5+448*x^6-1024*x^7-256*x^8+512*x^9 > which might look a bit nicer as > -1+5*z+10*z^2-20*z^3-15*z^4+21*z^5+7*z^6-8*z^7-z^8+z^9 > where z = 2*x. Thus 2*cos(pi/19) is an algebraic integer (which > is also obvious from the fact that it is exp(i pi/19) + exp(-i pi/19), > the sum of two roots of unity).
> Its conjugates are cos(3 pi/19), cos(5 pi/19), ..., cos(17 pi/19). > -- > Robert Israel isr...@math.MyUniversitysInitials.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada
Let M(z,a) be the minimal (monic) polynomial in z for the algebraic integer a. Then, re-writing Robert's expression above we have
> <isr...@math.MyUniversitysInitials.ca> wrote: > > master1729 <tommy1...@gmail.com> writes: > > > > what do you know about cos(pi/19) ?
> > > > nice expressions or properties ?
> > > algebraic order is 9.
> > Indeed, its minimal polynomial over the rationals is > > -1+10*x+40*x^2-160*x^3-240*x^4+672*x^5+448*x^6-1024*x^7-256*x^8+512*x^9 > > which might look a bit nicer as > > -1+5*z+10*z^2-20*z^3-15*z^4+21*z^5+7*z^6-8*z^7-z^8+z^9 > > where z = 2*x. Thus 2*cos(pi/19) is an algebraic integer (which > > is also obvious from the fact that it is exp(i pi/19) + exp(-i pi/19), > > the sum of two roots of unity).
> > Its conjugates are cos(3 pi/19), cos(5 pi/19), ..., cos(17 pi/19). > > -- > > Robert Israel isr...@math.MyUniversitysInitials.ca > > Department of Mathematics http://www.math.ubc.ca/~israel > > University of British Columbia Vancouver, BC, Canada
> Let M(z,a) be the minimal (monic) polynomial in z for the algebraic > integer a. Then, re-writing Robert's expression above we have
I don't understand why you get this. Apparently FunctionExpand simplifies Cos[Pi/k] iff it is constructable with compass and straightedge (i.e., iff k is a product of a power of 2 and distinct Fermat primes).
In particular, FunctionExpand[Cos[Pi/19]] returns Cos[Pi/19] (in Mathematica 6 and 7). What does it return on your system that evaluates numerically to 1.126478970802505 + 0.375464157076925*I ?
>> Indeed, its minimal polynomial over the rationals is >> -1+10*x+40*x^2-160*x^3-240*x^4+672*x^5+448*x^6-1024*x^7-256*x^8+512*x^9 >> which might look a bit nicer as >> -1+5*z+10*z^2-20*z^3-15*z^4+21*z^5+7*z^6-8*z^7-z^8+z^9 >> where z = 2*x. Thus 2*cos(pi/19) is an algebraic integer (which >> is also obvious from the fact that it is exp(i pi/19) + exp(-i pi/19), >> the sum of two roots of unity).
the fact that 2cos(pi/19)= exp(i pi/19) + exp(-i pi/19), =sum of two roots of unity prove only 2*cos(pi/19) is algebraic
so, there exists k in N^* such k*2*cos(pi/19) is algebraic integer
AP <marc.picher...@wanadoo.fr.invalid> writes: > > what do you know about cos(pi/19) ?
> >> algebraic order is 9.
> >> Indeed, its minimal polynomial over the rationals is > >> -1+10*x+40*x^2-160*x^3-240*x^4+672*x^5+448*x^6-1024*x^7-256*x^8+512*x^9 > >> which might look a bit nicer as > >> -1+5*z+10*z^2-20*z^3-15*z^4+21*z^5+7*z^6-8*z^7-z^8+z^9 > >> where z = 2*x. Thus 2*cos(pi/19) is an algebraic integer (which > >> is also obvious from the fact that it is exp(i pi/19) + exp(-i pi/19), > >> the sum of two roots of unity).
> the fact that 2cos(pi/19)= exp(i pi/19) + exp(-i pi/19), > =sum of two roots of unity > prove only 2*cos(pi/19) is algebraic
> so, there exists k in N^* such k*2*cos(pi/19) is algebraic integer
> but, why (without calculation) k=1? > thanks
exp(i pi/19) and exp(-i pi/19) are 38'th roots of unity, i.e. solutions of z^38 - 1 = 0. This is a monic polynomial with integer coefficients, so they are algebraic integers. The sum of two algebraic integers is an algebraic integer. -- Robert Israel isr...@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada
>> the fact that 2cos(pi/19)= exp(i pi/19) + exp(-i pi/19), >> =sum of two roots of unity >> prove only 2*cos(pi/19) is algebraic
>> so, there exists k in N^* such k*2*cos(pi/19) is algebraic integer
>> but, why (without calculation) k=1? >> thanks
>exp(i pi/19) and exp(-i pi/19) are 38'th roots of unity, i.e. solutions of >z^38 - 1 = 0. This is a monic polynomial with integer coefficients, so they >are algebraic integers. The sum of two algebraic integers is an algebraic >integer.
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