first of all, all finite fields have order a power of a prime, with p prime and . so it is not a finite field.

Alternatively, a field must be an abelian group under addition. There are 2 groups of order 10, and only one is an abelian group of order 10, the cyclic one. You can check with Sylow theorem or take my word for it. (If you are curious, the non abelian group of order 10 is the dihedral one). This is clearly not a field as and fields do not have zero divisors