<aside> 💡 Differences between fields and rings:
The main difference between a field and a ring in mathematics is that a field is an algebraic structure that contains two operations, addition and multiplication, and the multiplication operation is both commutative and associative. A ring, on the other hand, is an algebraic structure that contains two operations, addition and multiplication, but the multiplication operation is only associative. Additionally, a ring has the concept of an identity element for multiplication, which a field does not have.
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Both have mappings of addition and multiplication
We imply that all fields have addition and multiplication
char(F): is the smallest number for which the n fold sum of multiplicative neutral element 1 equals zero.
Basically like the order but for fields
rational numbers are a field, with addition, subtraction, multiplication and division
characteristic is 0, since there is no number n that adds 1 n times to get 0
The smallest field must have at least 2 elements, one is neutral element of addition and the other is neutral element of multiplication.
Remainder class sets Zn are commutative rings with units.
<aside> 💡 why is modular arithmetic groups commutative rings with units
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When modulus is prime, every remainder class has modular inverse.
We proved why every prime group means its cyclic and thus has a generator
EX: