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Unformatted text preview: More Examples of Fields Paul Skoufranis September 20, 2011 The purpose of this document is to introduce additional examples of fields (which will not be discussed in class due to time restrictions). In addition, several proofs will be given in order to provide students with additional examples and techniques. The notation of the field axioms will be denoted as in Appendix C of the text. In order to discuss our first field, we desire to provide a proof that √ 2 is irrational. Proposition) √ 2 is irrational. Proof : The following is an example of a proof by contradiction. Notice that either √ 2 is rational or √ 2 is irrational. Suppose to the contrary that √ 2 is rational. Then there exists integers a,b ∈ Z such that b 6 = 0, a and b have no common factors (meaning that no natural number larger than 1 divides both), and √ 2 = a b . By rearranging the equation and squaring both sides, we obtain that 2 b 2 = a 2 and thus a 2 must be an even integer. Since the square of an odd integer is odd and a 2 is even, a must be an even integer. Therefore there exists a c ∈ Z such that a = 2 c . By substituting 2 c for a in the above equation, we obtain that 2 b 2 = (2 c ) 2 = 4 c 2 so b 2 = 2 c 2 . Thus, by the same logic as before, b must be an even integer. Hence a and b are both even integers and thus have a common factor. However, we know that a and b have no common factors and thus we have a contradiction. Thus it is not possible that √ 2 is rational. Hence √ 2 must be irrational. With this in hand, we can now proceed to our first additional example of a field. Example) Let Q [ √ 2] := { a + b √ 2  a,b ∈ Q } . We claim that if + and · be the usual operations on R , then Q [ √ 2] is closed under + and · (meaning that if we add or multiply two elements of Q [ √ 2] then we get an element of Q [ √ 2]). If we can show this, then + and · might be operations that turn Q [ √ 2] into a field. To see that Q [ √ 2] is closed under + and · , we notice that if a,b,c,d ∈ Q then ( a + b √ 2) + ( c + d √ 2) = ( a + c ) + ( b + d ) √ 2 and ( a + b √ 2) · ( c + d √ 2) = ( ac + 2 bd ) + ( ad + bc ) √ 2 are both elements of Q [ √ 2] since a + c,b + d,ac + 2 bd,ad + bc ∈ Q whenever a,b,c,d ∈ Q . We claim that Q [ √ 2] is a field with these operation. To show this, we need to demonstrate the five field axioms are true on Q [ √ 2] with the usual addition and multiplication. It is clear that F1, F2, and F5 hold for Q [ √ 2] as these axioms are true on the larger field R with the same operators. Moreover, since 0 , 1 ∈ Q [ √ 2], F3 holds. Therefore, to complete the proof that Q [ √ 2] is a field, we need only demonstrate the existence of additive and multiplicative inverses in Q [ √ 2]....
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 Fall '08
 Kong,L
 Math, Addition, Prime number, Multiplicative inverse, additive inverse, zp

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