Take the homomorphism $\sigma: \zeta\to\zeta^5$ with the primitive root $\gamma=5$ modulo $p=23$. The primitive root $ 5 $ modulo $ 23 $ has order $ 22 $ modulo $ 23 $ and the $ 23^{\text{rd}} $ cyclotomic ring of integers has $ 22 $ homomorphisms. Therefore, we have the homomorphisms $ \sigma^k $, $ k=0,\dots,21 $, in this cyclotomic ring of integers.
The cyclotomic integer $ 2-\zeta $ has the norm $ 47\cdot 178,481 $ and the integer $ 178481 $ is coprime to the prime $ 47 $. Since a prime ideal lying over a prime $ q $ has a norm that is a power of the prime $ q $, the principal ideal $ \langle2-\zeta\rangle $ with the same norm $ 47\cdot 178,481 $ must factorize to one prime ideal $ P_{47} $ lying over the prime $ 47 $ times an ideal that is coprime to the ideal $ \langle47\rangle $. The difference
$$ {\sigma^j(2-\zeta)-\sigma^k(2-\zeta)}={(2-\zeta^{5^j})-(2-\zeta^{5^k})}={\zeta^{5^k}-\zeta^{5^j}}={\zeta^{5^k}(1-\zeta^{5^j-5^k})} $$
is a prime cyclotomic integer lying over the prime $ p $: we have
$$ \dfrac{x^p-1}{x-1}=1+x+x^2+\dots+x^{p-1}=\prod_{i=1}^{p-1}(x-\zeta^i). $$
Now plug in the value $ x=1 $. The left-hand side then yields the prime $ p=23 $ while the right-hand side has the factor $ 1-\zeta^{5^k-5^j} $ with $ \zeta^{23}=1 $. Therefore, the cyclotomic integers $ \sigma^j(2-\zeta) $ and $ \sigma^k(2-\zeta) $, $ j\not\equiv k\pmod{p} $ are coprime with $ p\nmid47\cdot 178,481 $: otherwise, some prime ideal would divide
- the ideals $ \langle\sigma^j(2-\zeta)\rangle $ and $ \langle\sigma^k(2-\zeta)\rangle $ and then their norm $ 47\cdot 178,481 $ plus
- the difference $ \langle\zeta^{5^k}(1-\zeta^{5^j-5^k})\rangle=\langle1-\zeta^{5^j-5^k}\rangle $ and its norm $ p $
though we have $ \gcd(47\cdot 178,481,p)=1 $.
We have the norm
$$ \prod_{i=1}^{p-1}\left(2-\zeta^i\right)=47\cdot 178,481 $$
so that the prime $ 47 $ splits to $ 22 $ conjugate prime ideals $ \sigma^iP_{47} $, $ i=0,\dots,21 $, lying over the prime $ 47 $ because each factor is divisible by exactly one conjugate prime ideal lying over the prime $ 47 $ and the factors are pairwise coprime with respect to the prime ideals lying over the prime $ 47 $ and the $ 22 $ conjugate prime ideals multiply up to the prime $ 47 $.
Assume that the prime ideal $ P_{47} $ is principal. Then there exists a cyclotomic integer $ p_{47}(\zeta) $ such that $ P_{47}=\langle p_{47}(\zeta)\rangle $. We have
$$ \langle47\rangle=\prod_{j=0}^{p-2}\left\langle\sigma^i p_{47}(\zeta)\right\rangle={\left\lbrace\prod_{j=0}^{\frac{p-1}{2}-1}\left\langle \sigma^{2i}p_{47}(\zeta)\right\rangle\right\rbrace\cdot\left\lbrace\prod_{j=0}^{\frac{p-1}{2}-1}\left\langle\sigma^{2i+1}p_{47}(\zeta)\right\rangle\right\rbrace}
={\left\langle\prod_{j=0}^{\frac{p-1}{2}-1}\sigma^{2i}p_{47}(\zeta)\right\rangle\cdot\left\langle\prod_{j=0}^{\frac{p-1}{2}-1}\sigma^{2i+1}p_{47}(\zeta)\right\rangle} $$
and the cyclotomic integers of the products in the last term are invariant under the homomorphism $ \sigma^2 $ because $ \sigma^{p-1}\equiv\text{id} $ so that these cyclotomic integers lie in the quadratic embedding. It is relatively easy to prove that the prime ideals lying over the prime $ 47 $ in the quadratic embedding are not principle so that the the prime ideal $ P_{47} $ cannot be principle in the cyclotomic ring of integers either.
Now take the cyclotomic integer $ g(\zeta)=-1-\zeta-\zeta^{2}+\zeta^{7}-\zeta^{15}-\zeta^{16} $. The ideal $ \langle g(\zeta)\rangle $ factorizes to $ (P_{47})^3 $. Because the class order of the prime ideal $ P_{47} $ divides the exponent $ n $ for any ideal $ (P_{47})^n$ that is principal, the class order of the prime ideal must be $ 3 $. Because the class order divides the order of the finite class group, the integer $ 3 $ divides the order of the class group.
The procedure of sending a prime ideal into the quadratic embedding is successful now and then for determining whether the prime ideal is not principle. However, the determination of the class order of prime ideals is more successful with Jacobi cyclotomic integers. Another way would be the determination of the class number. The class number of cyclotomic rings of integers is the product of two factors and one factor is relatively simple to compute. For the $ 23^{\text{rd}} $ cyclotomic ring of integers, the first factor is $ 3 $. The second factor is the class number of the real cyclotomic ring of integers and this factor can be determined to $ 1 $ by the Minkowski bound.
The method of sending the prime ideal into the quadratic embedding can be studied in more detail in chapter $ 6.4 $, here. The method by Jacobi cyclotomic integers can be taken from chapter $ 6.10 $ (or here) and the computation of the class number formula from chapter $ 14 $. The complete determination of the class group of the $ 23^{\text{rd}} $ cyclotomic ring of integers and many other cyclotomic rings of integers can be taken from chapter $ 21 $.