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Problem 1

# Define a field and give an example of i) an infinite field and ii) a finite field.

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A field is a set F with two binary operations, addition (+) and multiplication (·), satisfying specific algebraic properties, known as field axioms. An example of an infinite field is the set of rational numbers (Q) with the usual addition and multiplication operations. An example of a finite field is the set of integers modulo a prime number, such as $$\mathbb{Z}_5$$ with modulo 5 addition and multiplication operations.
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## Step 1: Definition of a field

A field is a set F with two binary operations, addition (+) and multiplication (·), such that for all elements a, b, and c in F, the following conditions, known as field axioms, are satisfied: 1. Associativity of addition: $$a+(b+c)=(a+b)+c$$ 2. Associativity of multiplication: $$a·(b·c)=(a·b)·c$$ 3. Commutativity of addition: $$a+b=b+a$$ 4. Commutativity of multiplication: $$a·b=b·a$$ 5. Existence of additive identity: There exists an element 0 in F such that $$a+0=a$$ 6. Existence of a multiplicative identity: There exists an element 1 in F, not equal to 0, such that $$a·1=a$$ 7. Existence of additive inverses: For every element a in F, there exists an element −a in F such that $$a+(−a)=0$$ 8. Existence of multiplicative inverses: For every element a in F, except for 0, there exists an element $$a^{−1}$$ in F such that $$a·a^{−1}=1$$ 9. Distributive law: $$a·(b+c)=a·b+a·c$$

## Step 2: Infinite field example

An infinite field is a field that has infinitely many elements. A well-known example of an infinite field is the set of rational numbers (denoted by Q) with the usual addition and multiplication operations. The set Q satisfies all the axioms needed for it to be a field.

## Step 3: Finite field example

A finite field is a field that has a finite number of elements. Finite fields are also known as Galois fields. Let's consider the set of integers modulo p, denoted by $$\mathbb{Z}_p$$, where p is a prime number. In this set, addition and multiplication are carried out with the modulo operation. For the finite field example, let's take the set of integers modulo 5, denoted by $$\mathbb{Z}_5$$, as our set. This set consists of the elements {0, 1, 2, 3, 4}. The addition and multiplication in $$\mathbb{Z}_5$$ are carried out with the modulo 5 operation, and this set satisfies all the axioms needed for it to be a field. In summary: 1. A field is a set F with two binary operations, addition and multiplication, that satisfy specific algebraic properties. 2. An example of an infinite field is the set of rational numbers Q with the usual addition and multiplication operations. 3. An example of a finite field is the set of integers modulo a prime number, such as $$\mathbb{Z}_5$$ with modulo 5 addition and multiplication operations.

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