Given $\mathbb{K}$ a field, what is the definition subfield of $\mathbb{K}$? Intuitively, I think a subfield as a subset of a field which itself is a field, in other words, satisfy the axioms that a field should satisfy. However though, my professor gave this definition, which I cannot see if two definitions are equivalent.
'$\mathbb{F}$ is said to be a subfield of $\mathbb{K}$ if $\mathbb{F}$ is a unital subring of $\mathbb{K}$'
I guess my question boils down to the question that 'would a unital subring of a field be itself a field?'
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$\begingroup$As I note in the comment, I expect that the intended definition is something like:
Definition. Let $\mathbb{F}$ and $\mathbb{K}$ be fields. Then $\mathbb{F}$ is a subfield of $\mathbb{K}$ if $\mathbb{F}$ is a unital subring of $\mathbb{K}$.
Your definition, on the other hand, is also missing a bit of context: it is not merely "a subset of a field which is itself a field", but rather "a subset of the field which is itself a field under the induced operations." That is, you need $\mathbb{F}\subseteq \mathbb{K}$, and you need the addition and multiplication of $\mathbb{F}$ that make it into a field to be the restrictions of the multiplication and addition of $\mathbb{K}$ to the subset.
I am also guessing that your professor does not require all rings to have a unity, nor do they require "subring" to share the same unity as the overring even when both have them. More on this below.
With those two provisos, the two definitions are equivalent. A field which is also a unital subring of $\mathbb{K}$ will be a field, will be contained in $\mathbb{K}$, and will have as its operations the restrictions of the operations of $\mathbb{K}$. Conversely, a subset of $\mathbb{K}$ which is a field under the induced operations must be a subring (abelian subgroup closed under multiplication), and must contain the identity because being a field, it must have a unity, and this unity must be the same as the one in $\mathbb{K}$, since in $\mathbb{K}$ the only solutions to $x^2=x$ are $x=0_{\mathbb{K}}$ and $x=1_{\mathbb{K}}$, and the only solutions in $\mathbb{F}$ are $0_{\mathbb{F}}=0_{\mathbb{K}}$ and $1_{\mathbb{F}}$, which must therefore be equal to $1_{\mathbb{K}}$. Hence, it is in fact a unital subring.
To answer the question asked, if you do not assume your $\mathbb{F}$ is in fact a field, then the definition is incorrect, as witnessed by the fact that $\mathbb{Z}$ is a unital subring of $\mathbb{Q}$.
As to why we ask that it be a unital subring, if you assume all rings are unital then subrings should always have the same unity as their overrings, because the multiplicative unity is actually a nullary operation that is part of the structure of your objects. However, if you do not assume all rings need to be unital rings, then subrings need not contain the same unity or a unity at all; so you want to specify that you want it to have the same unity as the overfield.
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