Degree of polynomials in finite fields

Question

How is the degree of a polynomial defined in a finite field? For example, is $deg(x^3) = 3$ or is $deg(x^3) = deg(x) = 1$ in $Z_3$?

What are the downsides to defining it formally vs relative to the actual field? In my opinion it would make more sense to definite it relative to the actual field (i.e after reducing the degree if its $>= p$) since it lets us think better about questions like how many roots the polynomial has (for e.g $x^3$ has at most 3 roots but it is equal to $x$ in $Z_3$ so it has at most 1 root). On the other hand, the polynomial $x^2$ is reducible but $1$ is not even though they are equal in $Z_3$.

Answer 1

Note that for the indeterminate $X$, we do not have $X^3=X$, even if $x^3=x$ is true for all $x\in \Bbb F_3$.

Abstractly, the polynomial ring (in a variable $X$ over a ring $R$) is defined as a ring (that is unique up to unique isomoprhism) denoted $R[X]$, together with a ring homomorphism $\iota\colon R\to R[X]$ and a map $i\colon \{X\}\to R[X]$, such that for all rings $A$ and ring homomorphisms $\phi\colon R\to A$ and maps $f\colon\{X\}\to A$, there exists a unique ring homomorphism $\eta\colon R[X]\to A$ with $\eta\circ \iota=\phi$ and $\eta\circ i=f$.

Constructively, we can obtain $R[X]$ by considering all formal sums of products of elements of $R$ and formal powers $X^n$, $n\in\Bbb N_0$, of $X$. (And the maps $\iota$ and $i$ mentioned above are the obvious ones). Note that we do construct something new and hence any properties that hold in $R$ are not guaranteed to still hold in the different $R[X]$. In your concrete example, $x^3=x$ holds in $\Bbb F_3$, but there is no reason to assume that it still holds in the bigger ring $\Bbb F_3[X]$.