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)

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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
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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.)
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4
Q

Describe the proof of the Uniqueness of zero concept

A
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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

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6
Q

What are the properties of arithmetic modulo

A
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7
Q

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

A
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