1) Fields Flashcards
1
Q
What is a field
A
A set K, together with functions -
* K × K → K : (a, b) → ab (Multiplication)
* K × K → K : (a, b) → a + b (Addition)
2
Q
What are the field axioms
A
- (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
3
Q
What is the Uniqueness of zero and the Uniqueness of one in a field
A
- 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.)
4
Q
Describe the proof of the Uniqueness of zero concept
A
5
Q
What does it mean if x dividies y
A
For x, y ∈ Z, we say that x divides y if there is t ∈ Z such that y = tx
6
Q
What are the properties of arithmetic modulo
A
7
Q
What are the proof of the properies of arthimetic modulo (Don’t need to know)
A