Subfields in Field of Characteristic p

For a field of characteristic $p$, show that it contains a subfield isomorphic to $\mathbb{Z}_p$.

In this problem, we explore the concept of subfields within fields of a specific characteristic. A field of characteristic $p$ has a fascinating structure, in which the integer field consists of integers modulo $p$. Understanding subfields in this context involves identifying how a smaller field structure can be mirrored within a larger one, which is foundational in abstract algebra and field theory.

When dealing with a field of characteristic $p$, one important property is the resemblance it bears to the ring of integers modulo $p$, denoted as $\mathbb{Z}_p$. This comes from the fact that within the field, the additive identity 0 and multiplicative identity 1 are repeated every $p$ elements, mimicking modular arithmetic. The key insight here is that the elements {0, 1, 2, ..., $p-1$} form a subfield under field operations, showcasing the fundamental theorem of finite fields.

Conceptually, this problem invokes the understanding of field extensions and the idea of an isomorphic structure. An isomorphism implies a bijective homomorphism, a deep algebraic insight. Grasping this concept allows a field of characteristic $p$ to be viewed as an extension of its subfield isomorphic to $\mathbb{Z}_p$. This underscores principles found in ring theory and finite field analysis, providing a groundwork for further exploration into the behavior of algebraic structures and their interrelations.