The left distributive law is x⋅(y+z)=xy+xz, and the right distributive law is (x+y)⋅z=xz+yz. Both laws must hold for all arbitrary elements x, y, and z in the field F.
Conditions: The laws apply to all elements x, y, z in the set F.; Multiplication distributes over addition.
The left distributive law is x⋅(y+z)=xy+xz, and the right distributive law is (x+y)⋅z=xz+yz. Both laws must hold for all arbitrary elements x, y, and z in the field F.
Conditions: The laws apply to all elements x, y, z in the set F.; Multiplication distributes over addition.
A nonempty set F is called a field if it satisfies three specific algebraic conditions: (i) F is an additive abelian group, (ii) the set of non-zero elements F - {0} is an abelian group with respect to multiplication, and (iii) both left and right distributive laws hold in F.
Conditions: F is a nonempty set.; The conditions must hold for all elements x, y, z in F.
A nonempty set F is called a field if it satisfies three specific algebraic conditions: (i) F is an additive abelian group, (ii) the set of non-zero elements F - {0} is an abelian group with respect to multiplication, and (iii) both left and right distributive laws hold in F.
Conditions: F is a nonempty set.; The conditions must hold for all elements x, y, z in F.