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Unformatted text preview: 7.3. ADJOINING ALGEBRAIC ELEMENTS TO A FIELD 323 Definition 7.3.3. An element ˛ in a field extension L of K is said to be algebraic over K if there is some polynomial p.x/ 2 KOExŁ such that p.˛/ D ; equivalently, there are elements k ;k 1 ;:::k n 2 K such that k C k 1 ˛ C C k n ˛ n D . Any element that is not algebraic is called transcendental . The field extension is called algebraic if every element of L is algebraic over K . In particular, the set of complex numbers that are algebraic over Q are called algebraic numbers and those that are transcendental over Q are called transcendental numbers . It is not difficult to see that there are (only) countably many algebraic numbers, and that, therefore, there are uncount- ably many transcendental numbers. Here is the argument: The rational numbers are countable, so for each natural number n , there are only count- ably many distinct polynomials with rational coefficients with degree no more than n . Consequently, there are only countably many polynomials....
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
- Fall '08