The Surreal nummbers, $\boldsymbol{No}$, are [according to Wikipidia](https://www.wikipedia.com/en/Surreal_number#/Arithmetic_closure) the biggest ordered field, and the Surrcomplex numbers are [the biggest field of characteristic 0](https://math.stackexchange.com/questions/2108944/surcomplex-numbers-and-the-largest-algebraically-closed-field). Biggest in the sense that every field (which is a set) is imbedded inside. 


In light of that I was wondering whether there is a nice characterization of the biggest field of characteristic $p > 0$?


As Wojowu noted, this isn't unique to the Surreals or to $On\_p$. The question then becomes is there a minimal such field? 



Biggest Field Of Characteristic $p$
Conway's [nimbers](https://en.wikipedia.org/wiki/Nimber) form an interesting answer for $p=2$. That every Field of characteristic $2$ embeds into it follows from the fact they form an algebraically closed Field and that they contain arbitrarily large sets of algebraically independent elements (which is immediate because the Field is proper-class-sized).


This has been generalized by DiMuro to arbitrary $p>0$ in his paper "On $\mathrm{On}\_p$", see [here](https://arxiv.org/abs/1108.0962). The resulting Field again is algebraically closed, and every characteristic $p$ Field embeds into it for the same reasons as above. The construction is more artificial than that of nimbers, but still.


By the way, let me make a remark: all the Fields mentioned in the thread, that is surreals, surcomplexes, nimbers and $\mathrm{On}\_p$, are not *unique* maximal fields, even up to isomorphism. Indeed, for instance, taking the field of rational functions $\mathrm{On}\_p(x)$, it is clearly not algebraically closed, hence not isomorphic to $\mathrm{On}\_p$, yet every characteristic $p$ Field will embed into it (since they embed into $\mathrm{On}\_p$). This does not change the answer, but I thought it's worth emphasizing.


To address the newly added question and the comment below, I do not know of any sense in which $\mathrm{On}\_p$ is minimal - it contains proper subFields which also have the property that other Fields embed into it. However, as the other answer says, this Field is unique (up to isomorphism) characteristic $p$ Field with this property which is algebraically closed (indeed, this is a unique algebraically closed Field in characteristic $p$). This also is true of surcomplexes in characteristic $0$. I don't know of a corresponding description for surreals - there are many real closed fields of given cardinality, but see Alec Rhea's comment.


