1) Fields

Question 1

What is a field

Answer 1

A set K, together with functions -
* K × K → K : (a, b) → ab (Multiplication)
* K × K → K : (a, b) → a + b (Addition)

Question 2

What are the field axioms

Answer 2

• (F1) For all a, b, c ∈ K, a + (b + c) = (a + b) + c.
• (F2) For all a, b ∈ K, a + b = b + a
• (F3) For all a ∈ K, 0 + a = a = a + 0
• (F4) For each a ∈ K, there is an element b ∈ K such that a + b = 0 = b + a.
• (F5) For all a, b, c ∈ K, a(bc) = (ab)c
• (F6) For all a, b ∈ K, ab = ba
• (F7) For all a ∈ K, 1a = a = a1
• (F8) For each element a ∈ K, if a ≠ 0, there is an element b ∈ K such that ba = 1 = ab
• (F9) For all a, b, c ∈ K, a(b + c) = ab + ac

Question 3

What is the Uniqueness of zero and the Uniqueness of one in a field

Answer 3

• If a ∈ K is such that x + a = x for all x ∈ K, then a = 0 (Uniqueness of zero.)
• If b ∈ K is such that xb = x for all x ∈ K, then b = 1 (Uniqueness of one.)

Question 4

Describe the proof of the Uniqueness of zero concept

Answer 4

Question 5

What does it mean if x dividies y

Answer 5

For x, y ∈ Z, we say that x divides y if there is t ∈ Z such that y = tx

Question 6

What are the properties of arithmetic modulo

Answer 6

Question 7

What are the proof of the properies of arthimetic modulo (Don’t need to know)

Answer 7