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2011-09-26 11:46:32 -0600 commented answer Simplification of expression with exponentials.

Excellent Thank you!!! I had resorted to what I call a "half clever work around". I found $e^{2\pi i/5}$ to 16 decimal places, and expand as a polynomial with $g$ as a variable, and in the last step I substituted the numerical value for $g$. I then had to manually go through and remove the terms of the form 1.2123123 e^{-16} which was a major pain. This is so much better, thanks!

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2011-09-26 11:44:32 -0600 marked best answer Simplification of expression with exponentials.

Coincidentally, something like this was the first thing I did with SAGE. Glad I can help you. It's really magic, but here's how it goes. Try to copy this in a SAGE notebook cell:

R=PolynomialRing(ZZ,'x,a,b,c,d')
x,a,b,c,d=R.gens()
S=R.quotient(1+x+x^2+x^3+x^4,'g,a,b,c,d')
g,a,b,c,d=S.gens()

Just to make sure, evaluate

g^5-1

(it should work out to 0). Then evaluate

(a+b*g+c*g^2+d*g^3)*(a+b*g^2+c*g^4+d*g)*(a+b*g^3+c*g+d*g^4)*(a+b*g^4+c*g^3+d*g^2)

and sure enough, the answer follows:

a^4 - a^3*b + a^2*b^2 - a*b^3 + b^4 - a^3*c + 2*a^2*b*c - 3*a*b^2*c -
b^3*c + a^2*c^2 + 2*a*b*c^2 + b^2*c^2 - a*c^3 - b*c^3 + c^4 - a^3*d +
2*a^2*b*d + 2*a*b^2*d - b^3*d - 3*a^2*c*d - a*b*c*d + 2*b^2*c*d +
2*a*c^2*d - 3*b*c^2*d - c^3*d + a^2*d^2 - 3*a*b*d^2 + b^2*d^2 +
2*a*c*d^2 + 2*b*c*d^2 + c^2*d^2 - a*d^3 - b*d^3 - c*d^3 + d^4
2011-09-26 11:44:31 -0600 received badge  Supporter (source)
2011-09-26 07:31:17 -0600 asked a question Simplification of expression with exponentials.

Hello,

I am looking at expanding the product:

(a+b*g+c*g^2+d*g^3)*(a+b*g^2+c*g^4+d*g)*(a+b*g^3+c*g+d*g^4)*(a+b*g^4+c*g^3+d*g^2)

where $g=e^{2\pi i /5}$ is a fifth root of unity. The problem is, after expanding, and telling sage the value of $g$, it does not collect terms according to the rule $1+g+g^2+g^3+g^4=0$ so there are over 5 times too many terms.

Is there any way to make this simplification automatic? For degrees higher then $5$ going through manually will not be an option. (It is almost faster to multiply everything myself then figure out the collection of the terms)

Thank you very much for any help on this, I have been stuck for the last while

2011-09-26 07:22:40 -0600 received badge  Editor (source)
2011-09-26 07:21:30 -0600 asked a question Number Fields and the Norm

Hello,

I am currently very confused about how to use [Number Fields in Sage].(http://www.sagemath.org/doc/reference/sage/rings/number_field/number_field.html). Things are not working.

Specifically I would like the norm of $\mathbb{Z}[\zeta_5]$ which should be a homogeneous polynomial of degree 4.

I tried:

K.<g> = NumberField(1+x+x^2+x^3+x^4)

a,b,c,d=var('a b c d')

(a+b*g+c*g^2+d*g^3).norm()

However that did not work.

I also tried:

K.<g>=CyclotomicField(5); K

a,b,c,d=var('a b c d')

(a+b*g+c*g^2+d*g^3).norm()

Instead of returning the norm in the Cyclotomic ring, this simply gave me the norm over the complex numbers, that is multiplication by the complex conjugate. These two things are very different, and the norm of the complex number is not at all correct. (It should be degree 4 not degree 2)

Any help is greatly appreciated, thank you