What are the left and right distributive laws in a field?

Q: What are the left and right distributive laws in a field?

The left distributive law is $x \cdot (y + z) = xy + xz$, and the right distributive law is $(x + y) \cdot z = xz + yz$. Both laws must hold for all arbitrary elements x, y, and z in the field F.

The left distributive law is

x \cdot (y + z) = xy + xz

, and the right distributive law is

(x + y) \cdot 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.

Reasoning, step by step

Identify the third condition of the field definition: Left and right distributive laws hold in F.
Write the left distributive law: $x \cdot (y + z) = xy + xz$ .
Write the right distributive law: $(x + y) \cdot z = xz + yz$ .
Verify that these equations connect the operations of addition and multiplication within the field.

Example

The video explicitly writes the equations on the paper: 'x.( $y+z$ )=xy+xz' and '( $x+y$ )z=xz+yz', while the speaker reads them out as the left and right distributive laws.

Common misconceptions

Assuming that only one distributive law is required if multiplication is commutative; the definition explicitly lists both.
Confusing the distributive laws with the associative laws of addition or multiplication.

Watch the explanation

YouTubeNormed Spaces(Lecture 1) Field(Definition)

4:00 – 5:05Watch this moment ↗

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