Product polynomial in $\mathbb{F}_7$ I need to compute the product polynomial $$(x^3+3x^2+3x+1)(x^4+4x^3+6x^2+4x+1)$$
when the coefficients are regarded as elements of the field $\mathbb{F}_7$.
I just want someone to explain to me what does it mean when a cofficient (let us take 3 for example) is in $\mathbb{F}_7$ ? I know that $\mathbb{F}_7= \mathbb{Z}/7\mathbb{Z}$

We use $3$ as shorthand for the coset $$7\mathbb{Z}+3=\{\ldots,-11,-4,3,10,17,\ldots\}$$ in $\mathbb{Z}/7\mathbb{Z}$.  Since $$\cdots=7\mathbb{Z}-11=7\mathbb{Z}-4=7\mathbb{Z}+3=3\mathbb{Z}+10=3\mathbb{Z}+17=\cdots,$$ when working in $\mathbb{Z}/7\mathbb{Z}$, we have $$\cdots=-11=-4=3=10=17=\cdots.$$
Practically, this means if $k$ arises as a coefficient, we can replace $k$ with $k \text{ mod } 7$.
