I have an understanding of that a subject is various from a ring or team. It involves the components of the area to be reachable by using + – * and /. I are unable to see how this functions in cryptography associated fields. This sort of as 2 ^ 255 – 19 primary field. How can we access all aspects in that integer-based subject with a divide operation?

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>I understand a field is different to a ring or group.

Learn what it is and how it differs from a ring or a group. No, not “I think I know what a field means”, but be able to define what a field is, give some examples, and be comfortable doing operations in them. For example, explain to yourself what GF( 3^2 ) looks like and how you would operate on it. Be familiar with polynomial rings.

> It requires the elements of the field can be reached via + – * and /.

Fields have those operations sure. Division is not what you think it is. It’s multiplication by the denominator’s multiplicative inverse. For example 2/5 mod 7 is the same as 2*3 mod 7 because (1/5 = 3 mod 7), because (3 = 5^{-1}) because (5*3 = 15 = 1 mod 7).

> I cannot see how this works in cryptography related fields.

Learn about finite fields first, pick up a book in abstract algebra. If you do not have a solid understanding of what these structures are, you will be lost from day 1. Blogposts and tutorials only give you hints as to what you need to learn, pick up a real math book please.

> Like 2^255 – 19 prime field.

Sure, you can define that field, but I’m guessing you’re looking at elliptic curves over that finite field. The points on an elliptic curve form a group with a surprising group operation. You can do interesting cryptography in that group because in most cases discrete log or diffie hellman is hard on that group.

> How can we reach all elements in that integer based field with divide operation?

Division doesn’t work like real division, division is multiplication by the inverse.