I wanted to ask what are field and field axioms? I have tried looking on Wikipedia and Wolfram But They are too are advanced and I cant a understand one bit.So please any help would be much appreciated. Also there is a question in my book :"What is the difference between Real and the Complex fields?" As I don't know anything about fields ,so I don't know the answer too and cant think of one as well ( as I repeat I don't know anything about fields),So please again any help would be much appreciated on understanding Fields and field axioms. THANKS VERY MUCH
A field is a set of numbers which satisfy some calculation rules. For the field the following rules hold:
- Addition has the neutral element 0
- Addition has an inverse (adding the negative part to a number)
- Addition is associative
- Multiplication has the neutral element 1
- Multiplication is always invertible (by division)
Multiplication is associative
And another important fact is that the distributive law holds!
Real fields are fields of real numbers while complex fields consist on complex numbers.
In addition to the axioms given by kryomaxim, a field also must satisfy (5).... Addition and multiplication are commutative , and (6).... $0 \ne 1$. If all of the requirements are met except commutative multiplication, it is called a Division Ring With Unit. (The "unit" is 1). There is no real number x that satisfies $x^2+1=0$ but there certainly is a complex number that does. The usual terminology for a field F which is a subset of a field G is that F is a sub-field of G.
(0) x + y ∈ F and x · y ∈ F for any x, y ∈ F (closure under addition and multiplication). (1) x + (y + z) = (x + y) + z and x · (y · z) = (x · y) · z for any x, y, z ∈ F (associativity of addition and multiplication). (2) x + y = y + x and x · y = y · x for any x, y ∈ F (commutativity of addition and multiplication). (3) x + 0 = x and x · 1 = x for all x ∈ F (0 and 1 are called the additive identity and the multiplicative identity, respectively). (4) For any x ∈ F, there is a w ∈ F such that x + w = 0 (existence of negatives). Moreover, if x ̸= 0, then there is also an r ∈ F such that x · r = 1 (existence of reciprocals). We denote w = −x and r = x −1 . 4 (5) x · (y + z) = x · y + x · z for any x, y, z ∈ F (distributivity of addition over multiplication).