How do we think of a field in the context of group theory?

Question

The definition of a field (in the context of group theory) that I've been taught is as follows:

"A field is defined as being a set $F$, combined with the binary operations $+$ and $\cdot$"

This (to me) implies that a field is a group given by $\{ F, * \}$, where $*$ denotes the 2 operations $+$ and $\cdot$

However, to be a group, it would have to have an identity element, but there cannot exist some $x \in F$ such that $f+x = f$ and $f \cdot x = e$. This would imply that I am incorrect in thinking that a field is like a group.

Answer 1

A group is a set equiped with just one operator that satisfies some axioms. A field is something else entirely since it has two operators. However if $(F,+,\cdot)$ is a field then $(F,+)$ is always an abelian group and so is $(F\setminus \{0\}, \cdot)$. These groups are typically called the additive group of the field and the multiplicative group of the field respectively.

Answer 2

Attached to a field $F$ are two groups, namely the additive group $(F,+)$ and the multiplicative $(F^{\times},\cdot)$. Every finite subgroup of the multiplicative group of a field is cyclic, i.e., of a particular easy structure. Furthermore, if we have a field extension $E\supset F$, which is Galois, then we obtain another interesting group, namely the Galois group $Gal(E/K)$. So there are certainly many relations between fields and groups. However, groups and fields are quite different things themselves, of course.