Question

definition of field?

Answer 1

Field? Like a soccer field?

Answer 2

In abstract algebra, a field is a nonzero commutative division ring, or equivalently a ring whose nonzero elements form an abelian group under multiplication.

Answer 3

https://en.wikipedia.org/wiki/Field_(mathematics)

Answer 4

what is the ring?

Answer 5

the (Mathematics) ring on the link?

Answer 6

a ring is a group with another operation that is associative and has identity and the later operation distributes over the group operation

Answer 7

example.

The real numbers

abelian group with respect to addition:
a+0=a=0+a for all a and 0 is real
a+b is a real number for all a,b
a+-a=0 for all real a
a+b=b+a

multiplication
a(bc)=(ac)b
1 is a real number
1*a=a*1=a for all a

distributes
a(b+c) = ab+ac for all ab

Answer 8

the first comment I left should say "abelian group"

Answer 9

Put it simply, ring is something with two operations, addition and multiplication. Ring addition follows the same properties of real number addition, namely,

- (a + b) + c = a + (b + c) for all a, b, c in R (+ is associative).
- a + b = b + a for all a, b in R (+ is commutative).
- There is an element 0 in R such that a + 0 = a for all a in R (0 is the additive identity).
- For each a in R there exists −a in R such that a + (−a) = 0 (−a is the additive inverse of a).
- a + b in R for all a, b in R. (Closure)

However, the multiplication is different in a ring compared to the real numbers. The multiplication in ring satisfies the following properties.

- (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c) for all a, b, c in R (⋅ is associative).
- There is an element 1 in R such that a ⋅ 1 = a and 1 ⋅ a = a for all a in R (1 is the multiplicative identity).
- a ⋅ b in R for all a, b in R. (Closure)

However, there is usually no multiplicative inverse for elements in ring, that is you cannot divide one element by another element in a ring. Furthermore, ring multiplication need not be commutative (a ⋅ b != b ⋅ a) and usually isn't.

Answer 10

A field is a ring satisfying addition properties. A field is something like real number (or should I say real number is a field). The addition axioms of a ring are unchanged for a field. However, the following multiplication axioms are added.

- For each a in R there exists b in R such that a ⋅ b = b ⋅ a = 1 (b is the multiplicative inverse of a).
- a ⋅ b = b ⋅ a for all a, b in R (⋅ is commutative).

Since there exist a multiplicative inverse for each element in a field, you can divide one element by another in a field.