Abstract: Finite (extension) fields have become central to many aspects of digital communication. Actual computation in such fields requires the equivalent of a prime number for the polynomial ring, i.e., a specific polynomial that is irreducible over the coefficient field (often GF(2)). This talk will begin with background on CRC's (Cyclic Redundancy Codes) which are used for error detection, including a brief description of the scheme which will be used in the coming GPS upgrade. The speaker will show that this subject leads naturally to a simple, yet common family of binary irreducible polynomials. Research at NSA on these has led to an extension of Swan's Theorem (1962) concerning the factor parity of trinomials.