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### How can I formally multiply two polynomials from different rings.

Suppose I take a polynomial from $R[x]$ say $x^2+5x$ and another polynomial from $R[y]$ say $y^3$. I want to formally multiply them and print $x^2y^3 +5xy^3$ as the output.

How can I do that?

Note: K is finite field of size 4 in 'a'

My efforts:

R_1=PolynomialRing(K,['z%s'%p for p in range(1,3)])
R_2=PolynomialRing(K,['x%s'%p for p in range(1,3)])

pp=R_1.random_element()
(a + 1)*z1^2 + (a)*z1*z2 + (a + 1)*z2 + (a)

qq=R_2.random_element()
x1*x2 + (a + 1)*x2^2 + x1 + 1


When I do pp*qq I get the following output

unsupported operand parent(s) for *: 'Multivariate Polynomial Ring in z1, z2 over Finite Field in a of size 2^2' and 'Multivariate Polynomial Ring in x1, x2 over Finite Field in a of size 2^2' 2 None

Suppose I take a polynomial from $R[x]$ $K[x]$, say $x^2+5x$ $x^2 + 5x$, and another polynomial from $R[y]$ $K[y]$, say $y^3$. $y^3$. I want to formally multiply them and print get $x^2y^3 +5xy^3$ + 5xy^3$as the output. output. How can I do that? Note: K K is the finite field of size 4 in 'a'a. My efforts: R_1=PolynomialRing(K,['z%s'%p sage: K.<a> = FiniteField(4) sage: R_1 = PolynomialRing(K, ['z%s' % p for p in range(1,3)]) R_2=PolynomialRing(K,['x%s'%p range(1, 3)]) sage: R_2 = PolynomialRing(K, ['x%s' % p for p in range(1,3)]) pp=R_1.random_element() range(1, 3)]) sage: pp = R_1.random_element() sage: pp (a + 1)*z1^2 + (a)*z1*z2 + (a + 1)*z2 + (a) qq=R_2.random_element() sage: qq = R_2.random_element() sage: qq x1*x2 + (a + 1)*x2^2 + x1 + 1  When I do pp*qq I get the following output unsupported operand parent(s) for *: 'Multivariate Polynomial Ring Ring in z1, z2 over Finite Field in a of size 2^2' and 'Multivariate Polynomial Polynomial Ring in x1, x2 over Finite Field in a of size 2^2' 3 None ### How can I formally multiply two polynomials from different rings. Suppose I take a polynomial from$K[x]$, say$x^2 + 5x$, and another polynomial from$K[y]$, say$y^3$. I want to formally multiply them and get$x^2y^3 + 5xy^3\$ as the output.

How can I do that?

Note: K is the finite field of size 4 in a.

My efforts:

sage: K.<a> = FiniteField(4)

sage: R_1 = PolynomialRing(K, ['z%s' % p for p in range(1, 3)])
sage: R_2 = PolynomialRing(K, ['x%s' % p for p in range(1, 3)])

sage: pp = R_1.random_element()
sage: pp
(a + 1)*z1^2 + (a)*z1*z2 + (a + 1)*z2 + (a)

sage: qq = R_2.random_element()
sage: qq
x1*x2 + (a + 1)*x2^2 + x1 + 1


When I do pp*qq I get the following output

unsupported operand parent(s) for *: 'Multivariate Polynomial Ring
in z1, z2 over Finite Field in a of size 2^2' and 'Multivariate Polynomial
Ring in x1, x2 over Finite Field in a of size 2^2'