We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 03 May 2011
Number theory has already been seen in earlier chapters of this book. It was used in the design of fast Fourier transform algorithms. We did make use of some ideas that only now will be proved. This chapter, which is a mathematical interlude, will develop the basic facts of number theory – some that were used earlier in the book and some that we may need later.
We also return to the study of fields to develop the topic of an extension field more fully. The structure of algebraic fields will be important to the construction of number theory transforms in Chapter 10 and also to the construction of some multidimensional convolution algorithms in Chapter 11 and for some multidimensional Fourier transform algorithms in Chapter 12.
Elementary number theory
Within the integer quotient ring Zq, some of the elements may be coprime to q, and, unless q is a prime, others will divide q. It is important to us to know how many elements there are of each type.
Definition 9.1.1 (Euler)The totient function, denoted ϕ(q), where q is an integer larger than one, is the number of nonzero elements in Zq that are coprime to q. For q equal to one, ϕ(q) = 1.
When q is a prime p, then all the nonzero elements of Zq are coprime to p, and so ϕ(p) = p – 1 whenever p is a prime.
To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.
