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Multiplication of elements of extension fields

Let

p=13
R = GF(p)

_.<v> = PolynomialRing(R)
R4.<v> = R.extension(v^4 - 2, 'v')
_.<w> = PolynomialRing(R4)
R16.<w> = R4.extension(w^4-v, 'w')


I need to compute

f1 = R16(1)
for i in range(34):
A=R4.random_element() #in my case, this is not random, its derived by an function
f1 = f1^2*A #this is not the whole code, but it displayes the problem


This element stays allways in R4. But the result i expect, is an element of R16.

So: How can I tell sage to use the R16 multiplication instead? I need a full degree 15 polynomial. If it is possible, in one variable (w, since w^4=v).

Multiplication of elements of extension fields

Let

p=13
R = GF(p)

_.<v> = PolynomialRing(R)
R4.<v> = R.extension(v^4 - 2, 'v')
_.<w> = PolynomialRing(R4)
R16.<w> = R4.extension(w^4-v, 'w')


I need to compute

f1 = R16(1)
for i in range(34):
A=R4.random_element() #in my case, this is not random, its derived by an function
f1 = f1^2*A #this is not the whole code, but it displayes the problem


This element stays allways in R4. But the result i expect, is an element of R16.

So: How can I tell sage to use the R16 multiplication instead? I need a full degree 15 polynomial. If it is possible, in one variable (w, since w^4=v).

Multiplication of elements of extension tower fields

Let

p=13
R = GF(p)

_.<v> = PolynomialRing(R)
R4.<v> = R.extension(v^4 - 2, 'v')
_.<w> = PolynomialRing(R4)
R16.<w> = R4.extension(w^4-v, 'w')


I need to compute

f1 = R16(1)
for i in range(34):
A=R4.random_element() #in my case, this is not random, its derived by an function
f1 = f1^2*A #this is not the whole code, but it displayes the problem


This element stays allways in R4. But the result i expect, is an element of R16.

So: How can I tell sage to use the R16 multiplication instead? I need a full degree 15 polynomial. If it is possible, in one variable (w, since w^4=v).

Furthermore: How can I tell Sage to print any coefficient in Hex?