Direct and inverse limits of finite fields

Question

Let $\left\{\mathbb{F}_{p^n}| n \in \mathbb{N}\right\}$ be a family of finite fields of characteristic $p$. Whenever $m$ divides $n$, we have the inclusion homomorphism $\mathbb{F}_{p^m} \to \mathbb{F}_{p^n}$. We also have the surjection $\mathbb{F}_{p^n} \to \mathbb{F}_{p^m}$ given by the norm map. This gives us two limits:

$$I = \lim_{\leftarrow} \mathbb{F}_{p^n}$$ $$D = \lim_{\rightarrow} \mathbb{F}_{p^n}$$

Now, I think both $I$ and $D$ are algebraically closed fields containing $\mathbb{F}_p$ - they're both characteristic $p$ and for every irreducible polynomial $f(x) \in \mathbb{F}_p[x]$ all it's roots are contained in $\mathbb{F}_{p^n}$ for some $n$, right? My question is, are they both THE algebraic closure of $\mathbb{F}_p$? I think an explicit isomorphism can be constructed using the universal property of direct and inverse limits, but I'm not sure how.

Answer 1

$I$ is not a field, or indeed a ring, since the norm maps are not ring homomorphisms.

Answer 2

@Lord Shark the Unknown, for information: it is true that the inverse limit of fields w.r.t. norms in general is not a field because the norm maps do not respect addition. However, in a very special - and very important - case, one can put a field structure on that limit. This is the construction of the so called "field of norms" of Fontaine and Wintenberger (*) around the 80's, which allowed a decisive breakthrough in the theory of $p$-adic Galois representations. Let me try to summarize that construction, which could perhaps be given as an exam problem on local fields.

One starts with a fixed prime $p$ and a local field $K$ with perfect residue field of characteristic $p$. Consider an infinite extension $L/K$ which is "arithmetically profinite", APF for short. For the precise definition of APF, see (*). Suffice it to say that a totally ramified Galois extension whose Galois group is a $p$-adic Lie group (e.g., ($\mathbf Z_p , +$)) is APF. The "field of norms" $X_K (L)= X(L/K)^* \cup$ {0} associated to an APF extension $L/K$ is a local field of characteristic $p$ which is constructed as follows. Its multiplicative group $X(L/K)^*$ is just the inverse limit w.r.t. norms of the multiplicative groups of the finite subextensions of $L/K$.

The main problem is to define addition in $X_K (L)$. For convenience, denote by $\mathcal E_{L/K}$ the filtered ordered set of finite subextensions of $L/K$. Let $K_0$ and $K_1$ resp. be the maximal unramified and tamely ramified subextensions of $L/K$ (which belong to $\mathcal E_{L/K}$). Let us define a valuation on $X_K (L)$. If $\alpha=(\alpha_E) \in X_K (L)$, $v_E(\alpha_E)$ for $E\in \mathcal E_{L/K}$ does not depend on $E$ and we can define $v(\alpha)=v_E(\alpha_E)$. Besides, for $x\in k_L$, the residue field of $L$, let $[x]$ the multiplicative representative of $x$ in $K_0$. For $ E\in \mathcal E_{L/K}$, let $x_E$ be the $[E:K_1]$-th root of [x] (this makes sense since $[E:K_1]$ is a power of $p$. For $E\subset E'$, one has $x_E = x_E'^{[E':E]}= N_{[E':E]} (x_E')$, and since $\mathcal E_{L/K_1}$ is cofinal in $\mathcal E_{L/K}$, the normic system $(x_E)_{E\in \mathcal E_{L/K_1}}$defines an element of $X_K (L)$, denoted $f_{L/K} (x)$. Using the APF condition, one can show the following theorem :

(i) Let $\alpha, \beta \in X_K (L)$. Then for all $E \in \mathcal E_{L/K}$, the norms $N_{E'/E}(\alpha_E'+\beta_E')$ (for $E\subset E'$) converge (following the filter of sections of $\mathcal E_{L/K}$) to an element $\gamma_E \in E$ and $\alpha + \beta = (\gamma_E)_{E\in \mathcal E_{L/K}}$ is an element of $X_K (L)$.

(ii) $X_K (L)$ is a local field of characteristic $p$, with $v(X_K (L))^* = \mathbf Z$. Moreover, $f_{L/K}$ is an embedding of the residue field $k_L$ into $X_K (L)$ which induces an isomorphism of $k_L$ onto the residue field of $X_K (L)$.

(*) J-M . Fontaine & J-P. Wintenberger, Le corps des normes de certaines extensions algébriques de corps locaux, CR Acad. Sci. Paris, 288 A (1979).