## Chinese Remainder Theorem (Pt. III)

**Point of Post: **This is a continuation of this post.

As a last number theoretic corollary (there are many more, but I don’t have the time to list them all) we prove the fundamental fact about the Euler Totient Function . Recall that is the function defined by

It is easy to calculate for certain numbers. For example, if for some prime we know that is NOT coprime to if and only if . So, the number of non-coprime numbers less than is since is true if and only if , and so there are choices. Thus, . It is then a basic fact of number theory that (i.e. the number of units in ). But, by the CRT we know that,as rings . But, recall that every ring isomorphism induces a group isomorphism on the group of units so that as groups we know . Recalling that the unit group of a product is the product group of the units we may finally conclude that . And so in particular

which, if you stare at it long enough says that

**Theorem: ***For every one has that*

All things considered, it’s pretty amazing what one little theorem can do!

**References:**

1. Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 2004. Print.

2. Bhattacharya, P. B., S. K. Jain, and S. R. Nagpaul. *Basic Abstract Algebra*. Cambridge [Cambridgeshire: Cambridge UP, 1986. Print.

[…] then that since and and commute we have, from basic group theory, that . And, since (where is Euler’s totient function). The conclusion […]

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