define what it means to say that F is a field.?

Answer 1

In brief, F is a field under the operations + and x if F is a commutative ring under + and x which has multiplicative inverses for all its nonzero elements.

Another definition: F is a field under the operations + and x if:
(1) F is an Abelian group under +
(2) The nonzero elements of F are an Abelian group under x
(3) F obeys the distributive law: for any a, b, c in F, we have (a + b)c = ac + bc and a(b + c) = ab + ac.

In more detail, F is a field under the operations + and x if:
(A1) Addition is associative: For any a, b, c in F, we have (a + b) + c = a + (b + c)
(A2) There is an element 0 in F such that for any a in F, 0 + a = a + 0 = a
(A3) For any element a in F, there is an element -a such that (a + -a) = (a + -a) = 0
(A4) Addition is commutative: For any a, b we have a + b = b + a
(M1) Multiplication is associative: For any a, b, c, we have (a x b) x c = a x (b x c).
(M2) There is an element 1 in F such that for any a in F, 1 x a = a x 1 = a.
(M3) For any nonzero element a in F, there is an element a^(-1) such that a x a^(-1) = a^(-1) x a = 1.
(M4) Multiplication is commutative: For any a, b we have a x b = b x a.
(D1) Left distributivity: For any a, b, c in F, it's true that a(b + c) = ab + ac.
(D2) Right distributivity: For any a, b, c in F, it's true that (a + b)c = ac + bc.

Examples of fields include the rational numbers; the real numbers; and the complex numbers. The set {0, 1} under the following operations is also a field:

+ defined by
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 0

x defined by
0 x 0 = 0
0 x 1 = 0
1 x 0 = 0
1 x 1 = 1

Answer 2

F is a letter, the first letter in FIELD, It is a whole lot easier to print F than it is to print F I E L D. So it means when you print F that F stands for field. This makes things a lot easier to jot down. Quicker too.