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lecture27

# lecture27 - 27 Fields from quotient rings In Lecture 26 we...

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27. Fields from quotient rings In Lecture 26 we have shown that the quotient ring R [ x ] / ( x 2 + 1) R [ x ] is isomorphic to C , so, in particular, it is a field, while R [ x ] / ( x 2 - 1) R [ x ] is not a field. The reason we did not get a field in the second case is clear: the polynomial x 2 - 1 is reducible, that is, has a non-trivial factorization x 2 - 1 = ( x - 1)( x +1), and we have seen in the proof from Example 3 that the existence of factorization ( x - 1)( x + 1) is what prevents R [ x ] / ( x 2 - 1) R [ x ] from being a field. On the other hand, x 2 + 1 is irreducible, although it is not clear how to deduce that R [ x ] / ( x 2 + 1) R [ x ] is a field just from the irreducibility of x 2 + 1. In this lecture we will settle the latter issue: we will show that if F is any field and p F [ x ] is a polynomial, the the quotient ring F [ x ] /pF [ x ] is a field ⇐⇒ p is irreducible. 27.1. Basic definitions. Definition. Let F be a field and p F [ x ] a polynomial with coefficients in F . Then p is called irreducible if (i) p is non-constant, or, equivalently, deg( p ) > 0; (ii) p does not have non-trivial factorizations, that is, p cannot be written as p = gh where g, h F [ x ] and both g and h are non-constant. Remark: Irreducible polynomials are direct counterparts of prime inte- gers. The convention not to consider constant polynomials as irreducible corresponds to the convention not to consider 1 as a prime number. As we will see shortly, the analogy between prime integers and irreducible polyno- mials goes well beyond the definition. Definition. Let F be a field and p F [ x ] a polynomial with coefficients in F . Then p is called monic if the leading coefficient of p is equal to 1. Next we define the greatest common divisor for polynomials. Definition. Let F be a field and f, g F [ x ] two polynomials. A polynomial d F [ x ] is called the greatest common divisor (gcd) of f and g if the following conditions hold: (i) d is monic (ii) d divides both f and g 1

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2 (iii) If h F [ x ] is another polynomial which divides both f and g , then h divides d .
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lecture27 - 27 Fields from quotient rings In Lecture 26 we...

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