What is Bourbaki's definition of subfield? or categorical definition of subfield?

Question

Let $F$ be a field.

Let $K$ be a subset of $F$ which is closed under two binary operations $+,\cdot$

Assume $(K,+,\cdot)$ is a field.

Is $K$ called a subfield of $F$ in Bourbaki's definition?

Or, should $K$ contain the unity $1_F$? that is $(K,\cdot)$ a submonoid of $(F,\cdot)$?

Answer 1

See Nicolas Bourbaki, Algebra I : Chapters 1-3 (1998), Page 103 :

Definition 3. Let $A$ be a ring. A subring of $A$ is any subset $B$ of $A$ which is a subgroup of $A$ under addition, which is stable under multiplication and which contains the unit of $A$.

The above conditions may be written as follows:

$$0 \in B, B+B \subset B, −B \subset B, B \circ B \subset B, 1 \in B.$$

If $B$ is a subring of $A$, it is given the addition and multiplication induced by those on $A$, which make it into a ring.

Examples. (1) Every subgroup of the additive group $\mathbb Z$ which contains $1$ is equal to $\mathbb Z$. Thus $\mathbb Z$ is the only subring of $\mathbb Z$.

See page 114 :

Let $K$ be a field. Every subring $L$ of $K$ which is a field is called a subfield of $K$.

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Note

See the History section of Ring :

Most or all books on algebra [e.g. B.L.Van der Waerden, (1930) Moderne Algebra. Teil I] up to around 1960 followed Noether's convention of not requiring a $1$. Starting in the 1960s, it became increasingly common to see books including the existence of $1$ in the definition of ring, especially in advanced books by notable authors such as Artin, Atiyah and MacDonald, Bourbaki [N.Bourbaki, (1998), Algebra I : Chapters 1-3], Eisenbud, and Lang. But even today, there remain many books that do not require a $1$.

Faced with this terminological ambiguity, some authors [...] have tried to adopt more precise terms [e.g. Joseph Rotman, Galois Theory (2nd ed.)] :

• rings with multiplicative identity: unital ring, unitary ring, ring with unity, ring with identity, or ring with $1$

• rings not requiring multiplicative identity: rng or pseudo-ring.