Theorem. The polynomial ring over a field is a Euclidean domain.

Proof. Let $K[X]$ be the polynomial ring over a field $K$ in the indeterminate $X$ . Since $K$ is an integral domain and any polynomial ring over integral domain is an integral domain, the ring $K[X]$ is an integral domain.

The degree $\nu(f)$ , defined for every $f$ in $K[X]$ except the zero polynomial, satisfies the requirements of a Euclidean valuation in $K[X]$ . In fact, the degrees of polynomials are non-negative integers. If $f$ and $g$ belong to $K[X]$ and the latter of them is not the zero polynomial, then, as is well known, the long division $f/g$ gives two unique polynomials $q$ and $r$ in $K[X]$ such that $$f = qg+r,$$ where $\nu(r) < \nu(g)$ or $r$ is the zero polynomial. The second property usually required for the Euclidean valuation, is justified by $$\nu(fg) = \nu(f)+\nu(g) \geqq \nu(f).$$

The theorem implies, similarly as in the ring $\mathbb{Z}$ of the integers, that one can perform in $K[X]$ a Euclid's algorithm which yields a greatest common divisor of two polynomials. Performing several consecutive Euclid's algorithms one obtains a gcd of many polynomials; such a gcd is always in the same polynomial ring $K[X]$ .

Let $d$ be a greatest common divisor of certain polynomials. Then apparently also $kd$ , where $k$ is any non-zero element of $K$ , is a gcd of the same polynomials. They do not have other gcd's than $kd$ , for if $d'$ is an arbitrary gcd of them, then $$d' \mid d \quad \mbox{and} \quad d \mid d',$$ i.e. $d$ and $d'$ are associates in the ring $K[X]$ and thus $d'$ is gotten from $d$ by multiplication by an element of the field $K$ . So we can write the

Corollary 1. The greatest common divisor of polynomials in the ring $K[X]$ is unique up to multiplication by a non-zero element of the field $K$ . The monic gcd of polynomials is unique.

If the monic gcd of two polynomials is 1, they may be called coprime.

Using the Euclid's algorithm as in $\mathbb{Z}$ , one can prove the

Corollary 2. If $f$ and $g$ are two non-zero polynomials in $K[X]$ , this ring contains such polynomials $u$ and $v$ that $$\gcd(f,\,g) = uf+vg$$ and especially, if $f$ and $g$ are coprime, then $u$ and $v$ may be chosen such that $uf+vg = 1$ .

Corollary 3. If a product of polynomials in $K[X]$ is divisible by an irreducible polynomial of $K[X]$ , then at least one factor of the product is divisible by the irreducible polynomial.

Corollary 4. A polynomial ring over a field is always a principal ideal domain.

Corollary 5. The factorisation of a non-zero polynomial, i.e. the presentation of the polynomial as product of irreducible polynomials, is unique up to constant factors in each polynomial ring $K[X]$ over a field $K$ containing the polynomial. Especially, $K[X]$ is a UFD.

Example. The factorisations of the trinomial $X^4-X^2-2$ into monic irreducible prime factors are
$(X^2-2)(X^2+1)$ in $\mathbb{Q}[X]$ ,
$(X^2-2)(X+i)(X-i)$ in $\mathbb{Q}(i)[X]$ ,
$(X+\sqrt{2})(X-\sqrt{2})(X^2+1)$ in $\mathbb{Q}(\sqrt{2})[X]$ ,
$(X+\sqrt{2})(X-\sqrt{2})(X+i)(X-i)$ in $\mathbb{Q}(\sqrt{2},\,i)[X]$ .