Why Euler's Totient function is important?

Question

First for it's importance in the field of abstract algebra:

• This function returns the cardinality (order) of the group $U(n)$ closed under modular multiplication.

• This function also returns the upper bound for the order of an arbitrary element in the $U(n)$.

And after that it's computational importance in modular arithmetic:

• Since residue exponention is not well-defined, reducing the exponent modulo $\varphi(m)$ is our only way out for simplyfing modular exponents due to the Euler-Totient theorem.

• With the same way above, if exponent of the element is relatively prime with the $\varphi(m)$, we can compute modular inverse of the exponent and with the help of it we can calculate modular roots. It is essential to RSA.

Are my points true? What i can add to this list?

Answer 1

A few things:

• You reduce many term polynomials, to just a few terms mod any value. (see here as applied to an integer, a less general form of polynomial).

• It shreds power towers down to size, with repeated use.

• It allows us to work in smaller numbers, rather than potentially trillion digit numbers.

• It allows us to pigeonhole principle coprime variable sets.

• You can generalize it to products of coprime arithmetic progressions. The product of the first 4 numbers in arithmetic progression 10k+9 have a product that is 1 mod 10 for example (193,401 for those wondering).

• Can be used to limit long division in finding the reptend length of fractions with coprime denominator in a given base.

• Cryptography ( forgot this)

  • etc.

Answer 2

Another reason from abstract algebra:

If $n$ is a positive integer then every group of order $n$ is cyclic if and only if $\gcd(n,\varphi(n))=1$.

You can read about the history and proof of this result here.