Intro to proofs mat102

Question

Construct a field with 5 elements. Draw the tables of addition and multiplication.

Hint 1. Let the elements of such a field be {0,1,a,b,c}

Hint 2. use the cancellation rules, you may use the axioms and claim/propositions about fields in the course notes. You do not need to write all the reasoning only drawing the tables for addition and multiplication is enough (but make sure to understand the reasoning).

Answer 1

To construct a field with 5 elements, we'll begin by defining the addition and multiplication tables for the given elements {0, 1, a, b, c} using the given hint and the axioms and properties of fields.

### The Field with 5 Elements {0, 1, a, b, c}:

#### Addition Table:

We start by defining the addition table based on the closure property, associativity, commutativity, identity element, and additive inverse properties.

| +  | 0  | 1  | a  | b  | c  |

| --- | --- | --- | --- | --- | --- |

| 0  | 0  | 1  | a  | b  | c  |

| 1  | 1  | a  | b  | c  | 0  |

| a  | a  | b  | c  | 0  | 1  |

| b  | b  | c  | 0  | 1  | a  |

| c  | c  | 0  | 1  | a  | b  |

#### Multiplication Table:

Next, we'll define the multiplication table based on the closure property, associativity, commutativity, identity element, and multiplicative inverse properties.

| ×  | 0  | 1  | a  | b  | c  |

| --- | --- | --- | --- | --- | --- |

| 0  | 0  | 0  | 0  | 0  | 0  |

| 1  | 0  | 1  | a  | b  | c  |

| a  | 0  | a  | b  | c  | 1  |

| b  | 0  | b  | c  | 1  | a  |

| c  | 0  | c  | 1  | a  | b  |

### Understanding the Reasoning:

The tables above are constructed based on the axioms and properties of fields. These tables satisfy the closure, associativity, commutativity, identity element, and inverse element properties for addition and multiplication, ensuring that the given set with the defined operations forms a field structure.

By understanding the reasoning behind the tables and the properties they adhere to, you can see how these tables demonstrate the behavior of addition and multiplication within a field with 5 elements.