This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: Fields: Question: is there a field on four elements? Lets 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 lets look at xy . Since xy 6 = x 0 (Proposition 2), we have xy 6 = 0. Since xy 6 = x 1, we have xy 6 =...
View Full Document