Thank you for your content. I did an undergrad in electronics many years ago and I hated the maths parts of it, but over the past year or so I have been flirting with it again and thanks to content like yours I am really starting to engage in a way that I didn't think was possible thirty years ago! Keep up the great work.
Well for starters a group has one binary operation and a field has two, so they definitely don't share the same axioms. For example, the integers are a group because we can add any two integers together and get another integer. However, what is the multiplicative inverse of 2 in the integers?
A group (G,○) has exactly 1 binary operation, while a field (F,●,★) has exactly 2 binary operations. A group does not have to be abelian, but a field is always commutative. A field is always a group, but a group is not necessarily a field.
(F,●,★) is a field if (F,●,★) is a commutative ring such that every elements in F\{0_F} is a unit. Since ●,★ are both binary operations, checking for clouser is absolutly unnecessary.
