showing that $n$th cyclotomic polynomial $\Phi_n(x)$ is irreducible over $\mathbb{Q}$

The proof which follows is the one provided by Gauss and it uses modular arithmetic in very ingenious way. We will summarize the results needed as follows:

1) For a given prime $ p$, the numbers $ 0, 1, 2, \ldots, (p - 1)$ form a finite field under the operations of addition and multiplication modulo $ p$.

2) Since these numbers form a field, say $ F_{p}$, we can talk about polynomials $ f(z)$ whose coefficients are in $ F_{p}$. The set of all such polynomials, say $ F_{p}[z]$, has the unique factorization property i.e. any such polynomial can be factored as a product of irreducible polynomials in $ F_{p}[z]$ in a unique way apart from the order of the factors. The proof is same as that used for normal polynomials with rational coefficients.

3) If $ f(z)$ is a polynomial in $ F_{p}[z]$ then $ \{f(z)\}^{p} = f(z^{p})$. This is true for constant polynomials by Fermat's theorem which says that $ a^{p} \equiv a\,\,\text{mod} (p)$. For higher degree polynomials this is achieved by induction by writing $ f(z) = az^{n} + g(z)$ and using binomial theorem to raise both sides to power $ p$. In so doing we only need to note that the binomial coefficients involved are divisible by $ p$.

The proof of irreducibility of $ \Phi_{n}(z)$ is done in two stages:

Stage 1:

Let $ \zeta$ be a primitive $ n^{th}$ root of unity and let $ f(z)$ be its minimal polynomial i.e. $ f(z)$ is monic (leading coefficient $ 1$), has rational coefficients and irreducible and $ f(\zeta) = 0$. Since $ \zeta$ is also a root of $ z^{n} - 1 = 0$, it follows that $ f(z)$ divides $ (z^{n} - 1)$ and by Gauss Lemma $ f(z)$ has integer coefficients as well. We now establish the following:

If $ p$ is any prime which does not divide $ n$ then $ \zeta^{p}$ is a root of $ f(z) = 0$.

Proof: Since $ \zeta$ is also a root of $ \Phi_{n}(z) = 0$ it follows that $ f(z)$ divides $ \Phi_{n}(z)$. Thus we have $ \Phi_{n}(z) = f(z)g(z)$ where $ g(z)$ is also monic and has integer coefficients (by Gauss Lemma). Since $ p$ is coprime to $ n$ it follows that $ \zeta^{p}$ is also a primitive $ n^{th}$ root. And therefore $ \Phi_{n}(\zeta^{p}) = 0$.

Assuming that $ \zeta^{p}$ is not a root of $ f(z) = 0$ (otherwise there is nothing to prove), we see that it must be a root of $ g(z) = 0$. Therefore $ \zeta$ is a root of $ g(z^{p}) = 0$. Since $ f(z)$ is the minimal polynomial of $ \zeta$, it follows that $ f(z)$ divides $ g(z^{p})$ so that $ g(z^{p}) = f(z)h(z)$ where $ h(z)$ is monic with integer coefficients. Also since $ \Phi_{n}(z)$ is a factor of $ (z^n - 1)$ so that we have $ z^{n} - 1 = \Phi_{n}(z)d(z)$ where $ d(z)$ is again monic with integer coefficients. We thus have the following equations: $$z^{n} - 1 = f(z)g(z)d(z)\tag{1}$$ $$g(z^{p}) = f(z)h(z)\tag{2}$$ We now apply the modulo $ p$ operation to each of the equations above, i.e. we replace each coefficient in the polynomials involved with its remainder when it is divided by $ p$. The resulting polynomials are all in $ F_{p}[z]$ and we will use the same letters to denote them. So the above equations are now to be interpreted as relations between some polynomials in $ F_{p}[z]$. The equation $(2)$ can now be equivalently written as $$\{g(z)\}^{p} = f(z)h(z)\tag{3}$$ Let $ k(z)$ in $ F_{p}[z]$ be an irreducible factor of $ f(z)$. Then from the above equation $(3)$, $ k(z)$ divides $ \{g(z)\}^{p}$ and so divides $ g(z)$. Thus from equation $(1)$ the polynomial $ \{k(z)\}^{2}$ divides $ (z^{n} - 1)$. Thus $ (z^n - 1)$ has repeated factors and therefore $ (z^{n} - 1)$ and its derivative $ nz^{n - 1}$ must have a common factor. Since $ n$ is coprime to $ p$, therefore the derivative $ nz^{n - 1}$ is non-zero polynomial and it clearly does not have any common factor with $ (z^{n} - 1)$.

We have reached a contradiction and therefore the initial assumption that $ \zeta^{p}$ is not the root of $ f(z) = 0$ is wrong. The result is now proved.

Stage 2:

We have thus established that if $ f(z)$ is the minimal polynomial for any primitive $ n^{th}$ root of unity then for any prime $ p$ not dividing $ n$, $ \zeta^{p}$ (which is again a primitive $ n^{th}$ root) is also a root of $ f(z) = 0$. And since $ f(z)$ is irreducible and monic, it will act as minimal polynomial for the primitive root $ \zeta^{p}$.

The same logic can be applied repeatedly and we will get the result that $ f(z)$ is the minimal polynomial for $ \zeta^{p_{1}p_{2}\ldots p_{m}}$ where $ p_{1}, p_{2}, \ldots, p_{m}$ are any primes not dividing $ n$. It follows that $ \zeta^{k}$ where $ k$ is coprime to $ n$ is also a root of $ f(z)$. Thus all the primitive $ n^{th}$ roots of unity are roots of $ f(z) = 0$. Hence $ \Phi_{n}(z)$ divides $ f(z)$. Since $ f(z)$ is irreducible it follows that $ f(z) = \Phi_{n}(z)$ (both $ f(z)$ and $ \Phi_{n}(z)$ are monic). We thus have established that $ \Phi_{n}(z)$ is irreducible.

Update: User Vik78 points out in comments that Gauss had proved the irreducibility of $\Phi_{n}(z)$ for prime values of $n$ only and it was Dedekind who established it for non-prime values of $n$. Note that the problem is far simpler when $n$ is prime (one easy proof is via Eisenstein's criterion of irreducibility) and the crux of the argument presented above is essentially due to Gauss. I came to know of this proof from the wonderful book Galois Theory of Algebraic Equations by Jean Pierre Tignol.

Tignol mentions in his book that Gauss proved the irreducibility of $\Phi_{n}(z)$ over $\mathbb{Q}$ for non-prime values of $n$ in 1808 (see section 12.6 titled Irreducibility of cyclotomic polynomials on page 196 in the book mentioned in last paragraph). Dedekind on the other hand proved an even more powerful theorem:

Theorem: Let $\zeta_{m}$ denote a primitive $m^{\text{th}}$ root of unity. If $m, n$ are relatively prime then $\Phi_{n}(z)$ is irreducible over the field $\mathbb{Q}(\zeta_{m})$.

And Gauss used this result implicitly (thinking that it could be proved in similar manner as irreducibility of $\Phi_{n}(z)$ over $\mathbb{Q}$) in obtaining solution to $\Phi_{n}(z) = 0$ via radicals.

I will try to rephrase the essence of answer by Paramanand Singh so as to somewhat better isolate the arguments used (though probably less faithful to what Gauss wrote*), for the sake of transparency.

The starting point is that $\def\Z{\Bbb Z}\Phi_n\in\Z[X]$ are inductively defined for $n>0$ by the recurrence relation $\prod_{k\mid n}\Phi_k=X^n-1$ (just like the numbers $\phi(n)$, which ar their degrees, are defined by $\sum_{k\mid n}\phi_k=n$); one can compute $\Phi_n$ in $\def\Q{\Bbb Q}\Q[X]$ by polynomial exact division starting from $X^n-1$, and since (by induction) all divisions are by monic polynomials with integer coefficients, $\Phi_n$ is monic and has integer coefficients.

If $\Phi_n$ were reducible over$~\Q$, this would partition the primitive $n$-th roots of unity (in $\def\C{\Bbb C}\C$) into more than one subset, each characterised as the roots of a different rational polynomial. To show that this is impossible, one argues that for any irreducible factor $F$ of $\Phi_n$ in $\Q[X]$, and for any prime$~p$ not dividing$~n$, the set of roots of $F$ is closed under the operation $\zeta\mapsto\zeta^p$. Since (by consideration of the cyclic group of $n$-th roots of unity) this operation maps primitive $n$-th roots to primitive $n$-th roots, and compositions of such operations for different primes$~p$ allow going from any one of the $\phi(n)$ primitive $n$-th roots to any other, this will show that $F$ has $\phi(n)$ distinct roots in$~\C$, and therefore equals $\Phi_n$.

Now fix such $F$ and $p$. The operation $\zeta\mapsto\zeta^p$ in $\C$ on roots gives rise to an operation on irreducible polynomials in $\Q[X]$: the minimal polynomial$~G$ over$~\Q$ of the image of $X^p$ in the field $\Q[X]/(F)$ is a monic irreducible polynomial in $\Q[X]$ determined by$~F$, and by construction $F$ divides $G[X^p]$, the result of substituting $X^p$ for $X$ in $G$. Whenever $\zeta\in\C$ is a root of$~F$ it is clear that $\zeta^p$ is a root of$~G$, and since $\deg G\leq[\Q[X]/(F):\Q]=\deg F$ one gets all roots of $G$ this way. One must show that $F=G$.

A key point is that $F$ and $G$, both of which divide $\Phi_n$ in $\Q[X]$ since their (distinct) roots in$~\C$ are among the roots of the latter, have integer coefficients. This is because of Gauss's lemma stating that the product of primitive polynomials in $\Z[X]$ (i.e., those not divsible by any prime number) is again primitive. (A detailed argument goes: $F\in\Q[X]$ is a monic divisor of the monic $\Phi_n\in\Z[X]$, so the quotient $Q\in\Q[X]$ with $FQ=\Phi_n$ is monic; then some positive integer multiples $aF,bQ\in\Z[X]$ are primitive (namely the minimal such multiples with all integer coefficients), and by the lemma $aFbQ=ab\Phi_n$ is primitive, but since $\Phi_n\in\Z[X]$ this forces $a=b=1$.)

So one can apply modular reduction $\def\Fp{\Bbb F_p}\Z[X]\to\Fp[X]$ to our polynomials, which I shall write $P\mapsto\overline P$. Being a morphism of rings, it preserves divisibility of polynomials. An important point is that $\overline{X^n-1}$ is still square-free over$~\Fp$ (it has no multiple roots in an extension field), since it is coprime with its derivative $\overline{nX^{n-1}}$ (as $\overline n\neq0$ in$~\Fp$ by hypothesis). Now if $F$ and $G$, which are both irreducible factors of$~X^n-1$, were distinct, then $\def\oF{\overline F}\oF$ and $\def\oG{\overline G}\oG$ would (also) be coprime in$~\Fp[X]$. We will show this to be false.

Since in any ring of characteristic$~p$ the map $\eta:x\mapsto x^p$ is a morphism of rings (called the Frobenius endomorphism), and $\eta$ fixes all elements of$~\Fp$, one has in $\Fp[X]$ that $\oG^p=\eta(\oG)=\oG[X^p]$, which is also clearly equal to $\overline{G[X^p]}$ and therefore divisible by $\oF$. This contradicts that $\oF$ and $\oG$ are coprime, and this contradiction finishes the proof.

*Googling around a bit I found evidence that the result is not due to Gauss at all for general $n$, but only for prime$~n$ (where $\Phi_n=1+X+\ldots,X^{n-1}$). See this note and this one. Apparently the general case was first proved by Dedekind, and the simplification leading to above proof is due to van der Waerden.

According to the Book Algebra of Serge Lang, the "fact that $\Phi_n$ is an irreducible polynomial of degree $\varphi(n)$ in the ring $\mathbb{Z}[x]$ is a nontrivial result due to Gauss". So there is no short answer to your question.

Anyway, just open your favorite book on abstract algebra, and find the proof there. (If not, maybe it's time to change your "favorite" book...)

Also, google brought up this document where several different proofs are given.

There is also a non-elementary proof that perhaps is more explanatory than the "elementary" argument, using some (but not too much) algebraic number theory, and primes in arithmetic progressions. To prove that $\mathbb Q(\zeta_{p^kN})$ has the expected degree over $\mathbb Q$ (where $p$ does not divide $N$), it suffices to do an induction, namely, that $\mathbb Q(\zeta_{p^kN})$ has the expected degree over $\mathbb Q(\zeta_N)$. (Good so far!) For this, it suffices to find a prime $q$ so that $\mathbb Q_q$ already contains a primitive $N$th root of unity, so that $\mathbb Q_q(\zeta_N)$ just collapses to $\mathbb Q_q$, but at the same time so that $\mathbb Q_q(\zeta_{p^kN})=\mathbb Q_q(\zeta_{p^k})$ has the expected degree over $\mathbb Q_q(\zeta_N)=\mathbb Q_q$. By Dirichlet's theorem on primes in an arithmetic progress, there is a prime $q$ such that $q=1\mod N$, while $q$ is congruent to a primitive root mod $p^k$. Then the finite field $\mathbb F_q$ contains a primitive $N$th root of unity, which lifts to a primitive $N$th root of unity in $\mathbb Q_q$. Meanwhile, since the degree of an extension is at least the residue class field extension degree) $\mathbb Q_q(\zeta_{p^k})$ is of the expected (maximal possible) degree, namely, $(p-1)p^{k-1}$. (Unless I've maintained old typos or created new ones... Hopefully the verbal description makes things more robust...)

(EDIT: Thanks to @M.A.Sarkar for gentle prodding to make corrections above.)

I think it's worthy to mention the proof in Washington's cyclotomic fields book, Proposition 2.4.

This proof is really easy to remember when the basic ramification theory is established, but there is also something nontrivial. That is, if $K$ is a number field, then $|d(K)|>1$ if $K\neq \mathbb{Q}$, where $d(K)$ is the discriminant of $K$. This is due to Minkowski.

Since we know the irreducibility for prime power cases. Then the irreducibility is equivalent to $K:=\mathbb{Q}(\zeta_n)\cap \mathbb{Q}(\zeta_m)=\mathbb{Q}$ when $(n,m)=1$. By consider ramify condition, we know $K$ is unramify for every prime numbers, then by the above theorem, $K=\mathbb{Q}$.