Hello, I have a polynomianl coefficients in Vector1 = [[4,7],[3,1]]. I cant add polynominal in Matrix. In result i need Matrix =Gf(p^n),1,n) = {4x+7,3x+1}. But i can do it in manually. Vector2=Matrix(Gf(p^n),1,n)) Vector2[0,0]=Vector1 Vector2[0,1]=Vector1 And in result i have need matrix. how do this right.

P.S. Sorry, English it is not my native language.

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Please describe mathematically what is needed, then we can see how to come together. Simple questions: We want a $2\times 2$ matrix with entries in the polynomial ring over some finite field? Please use then LaTeX to declare it explicitly. Please define $n$, better, give an example with a fixed $n$.

Is this a correct rephrasing of your question?

I have a list of polynomial coefficients stored as Vector1 = [[4, 7], [3, 1]].

I need to transform that into a matrix over GF(p^n) which would look like {4*x+7, 3*x+1}, and that could be added to another matrix.

Stored as above, adding vectors does not work as expected.

But i can do it in manually.

Vector2 = Matrix(Gf(p^n), 1, n)
Vector2[0, 0] = Vector1
Vector2[0, 1] = Vector1


And in result I need a matrix. How do this right.

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Hints:

• one can easily convert a list of numbers into a polynomial.
• one can easily convert a polynomial into a finite field element.

Here is an example which hopefully addresses the question.

Define a number field:

sage: p = 29
sage: n = 2
sage: F.<x> = GF(p^n)


Define a polynomial ring over the integers.

sage: R = PolynomialRing(ZZ, 't')


Now, convert a list of coefficients (from degree 0 to degree n) into a polynomial.

sage: u = [7, 4] # careful: start from degree 0 coefficient
sage: g = R(u)
sage: g
4*t + 7


Now, turn this polynomial into a finite field element:

sage: z = F(g)
sage: z
4*x + 7


Combining this (again, careful with order of coefficients), define lists of lists of coefficients:

sage: u1 = [[7, 4], [1, 3]]
sage: u2 = [[1, 3], [2, 5]]


turn them into vectors:

sage: v1 = vector(F, [R(u) for u in u1])
sage: v2 = vector(F, [R(u) for u in u2])


and now you can add them:

sage: v1 + v2
(7*x + 8, 8*x + 3)
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