As far as I know, "the bigger one" has no canonical definition in a finite field. At least, there is no total order compatible with addition, that is such that $a\le b$ and $a'\le b'$ implies $a+a'\le b+b'$ (as we have in fields $\mathbb Q$ and $\mathbb R$).
Thus, we first need to select some ordering criteria. A most natural candidate definition uses a mapping of the field $\mathbb F_{p^k}$ to $[0,p^k)$ (the non-negative integers less than $p^k$) as follows: consider the polynomial representing a field element as with all the terms of the polynomial in $[0,p)$, and evaluate that polynomial‡ in $\mathbb N$ at point $p$. We then build a total order on the field from the canonical total order in $\mathbb N$.
Once we have defined "the bigger one" one way or another, that defines a $\max$ function from $\mathbb F_{p^k}\times\mathbb F_{p^k}$ to $\mathbb F_{p^k}$. And there is a generic method to implement any function with such domain using an arithmetic circuit, as a polynomial of the two inputs:
- for each $J$ in the field we can build a Lagrange polynomials $P_J(X)$ with coefficients in the field evaluating to $1$ if $X=J$ and $0$ otherwise, per$$P_J(X)=-\prod_{I\ne J}(X-I)$$
- and now $\max(X,Y)$ can be built as a polynomial, per$$\max(X,Y)=\sum_{I,J}\bigl(\max(I,J)\,P_I(X)\,P_J(Y)\bigr)$$
This polynomial can be developed into the sum of terms $U_{i,j}\,X^i\,Y^j$ with $i,j\in[0,p^k)$ where $U_{i,j}$, $X$ and $Y$ are elements of the field $\mathbb F_{p^k}$. At least $U_{0,0}$ is zero, and $U_{i,j}=U_{j,i}$.
Example: For field $\mathbb F_7$ and the above definition, $\max(x,y)$ is $(3 x^6 y^2+3 x^6 y+x^5 y^3+6 x^5 y+4 x^4 y^4+3 x^4 y^2+x^3 y^5+3 x^3 y^3+3 x^3 y+3 x^2 y^6+3 x^2 y^4+x^2 y^2+3 x y^6+6 x y^5+3 x y^3+2 x y+x+y)\bmod 7$.
I wonder if there are much simpler expressions of that polynomial, and/or if one even simpler can be made by loosening the order.
[Clarified] I'm willing to trust the other answer+comment's statement that it's provably impossible to craft an arithmetic circuit of size polynomial in $k\log p$ that implements $\max(X,Y)$ as defined above.
‡ For $k=1$, this is comparison in $[0,p)$. For $p=2$, this is unsigned binary conversion then comparison in $[0,2^k)$. More generally this is base-$p$ conversion then comparison in $[0,p^k)$.
