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View polynomial over a field extension as a polynomial in the basis elements of the extension

asked 2021-04-30 02:45:50 +0200

seagriffins gravatar image

I have a finite field extension L/K, with basis 1, z, ..., z^n, and a polynomial f in L[x,y]. I would like to regard f as a polynomial in z and obtain its K[x,y] coefficients as such. How can I do this?

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Can you give us some concrete example ?

Maybe have a look at the documentation ?

Emmanuel Charpentier gravatar imageEmmanuel Charpentier ( 2021-04-30 09:15:20 +0200 )edit

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answered 2021-04-30 11:41:28 +0200

rburing gravatar image

Example setup:

K = QQ
X = polygen(K)
L.<z> = K.extension(X^2 - 2)
R.<x,y> = PolynomialRing(L)
f = (x+z*y)^3

An element of $L$ can be viewed as a vector over $K$ of length $n$. Apply this to each coefficient of $f \in L[x,y]$ and multiply the vector by the respective monomial (considered as an element of $K[x,y]$) to get a vector over $K[x,y]$. Then add all those vectors:

sage: f
x^3 + (3*z)*x^2*y + 6*x*y^2 + (2*z)*y^3
sage: sum(c.vector()*m.change_ring(K) for c,m in f)
(x^3 + 6*x*y^2, 3*x^2*y + 2*y^3)
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Asked: 2021-04-30 02:45:50 +0200

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Last updated: Apr 30 '21