# Fields - Fields Question is there a field on four elements...

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Unformatted text preview: Fields: Question: is there a field on four elements? Let’s call the four elements in the set S : 0 , 1 ,x,y . To discover what the multiplication table would have to look like, the follow- ing observations will be useful: 1. x 0 = 0 for all elements in the set Proof: Let y be in the set S , then y = y + 0 ⇒ xy = x ( y + 0) ⇒ xy = xy + x ⇒ 0 = x 2. If y,z are distinct and x is nonzero, then xy and xz are distinct. Proof: Again, we will prove an equivalent statement: If xy = xz then y = z or x = 0. So assume xy = xz . If x = 0 we are done because the conclusion holds. Otherwise x has an inverse x- 1 xy = xz ⇒ x- 1 xy = x- 1 xz ⇒ y = z To put this into context of the table, every element in the set must appear in every row/column of nonzero elements. Now we begin to fill in the multiplication table using Proposition 1 and the axiom 1 x = x : · 1 x y 1 1 x y x x y y 1 Next let’s look at xy . Since xy 6 = x 0 (Proposition 2), we have xy 6 = 0. Since xy 6 = x 1, we have xy 6 =...
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Fields - Fields Question is there a field on four elements...

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