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 $Z_3$.

I think you are confusing polynomials in $R[X]$ and the functions on $R$ they define. See, for example, this thread. — Alex Macedo, Jan 18 '20 at 08:25
Also, $x^2 \neq 1$ as functions on $\mathbf F_3$ since $1 \neq 0 = 0^2$ — Alex Macedo, Jan 18 '20 at 08:28
Good job finding that related thread @Alex. Saad, see also this oldie for more discussion on this theme. All: this might be a duplicate of one of those, but I am undecided. — Jyrki Lahtonen, Jan 18 '20 at 18:53
For much further discussion see Why are polynomials defined to be “formal”? and Axiomatic approach to polynomials? — Bill Dubuque, Jan 18 '20 at 21:22

score 3 · Accepted Answer · answered Jan 18 '20 at 08:17

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 $\Bbb F_3[X]$.

Degree of polynomials in finite fields

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