1 fields

Set of elements with defined addition and multiplication: (\mathbb{F}, +, \cdot) It has to be comply the following rules:

• Associativity of addition and multiplication: a + (b + c) = (a + b) + c, and a \cdot (b \cdot c) = (a \cdot b) \cdot c
• Commutativity of addition and multiplication: a + b = b + a, and a \cdot b = b \cdot a
• Additive and multiplicative identity: there exist two different elements \mathbf{0} and \mathbf{1} in \mathbb{F} such that a + \mathbf{0} = a and a \cdot \mathbf{1} = a
• Additive inverses: for every a in \mathbb{F}, there exists an element in \mathbb{F}, denoted -a, called the additive inverse of a, such that a + (-a) = 0
• Multiplicative inverses: for every a \ne 0 in \mathbb{F}, there exists an element in \mathbb{F}, denoted by a^{-1} or \frac{1}{a}, called the multiplicative inverse of a, such that a \cdot a^{-1} = 1
• Distributivity of multiplication over addition: a \cdot (b + c) = (a \cdot b) + (a \cdot c)

1.0.1 groups vs rings vs fields

• group has a defined addition and subtraction
• ring has a defined addition, subtraction, and multiplication
• field has a defined addition, subtraction, multiplication, and division (however addition and multiplication is enough as the other ones can be defined using these)