College Algebra Exam Review 313

College Algebra Exam Review 313 - 7.3. ADJOINING ALGEBRAIC...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

Ask a homework question - tutors are online