Intereting Posts

How many groups of order $2058$ are there?
How did we find the solution?
Derivation of weak form for variational problem
prove that $\tan(\alpha+\beta) = 2ac/(a^2-c^2)$
Find: $ \int^1_0 \frac{\ln(1+x)}{x}dx$
Prove that $H$ is a subgroup of an abelian group $G$
What could be better than base 10?
Calculate: $\lim\limits_{x \to \infty}\left(\frac{x^2+2x+3}{x^2+x+1} \right)^x$
Can a continuous surjection between compacts behave bad wrt Borel probability?
Integrate complex conjugate
Determinant of the transpose via exterior products
Showing there exists infinite $n$ such that $n! + 1$ is divisible by atleast two distinct primes
Angle preserving linear maps
Why not just study the consequences of Hausdorff axiom? What do statements like, “The arbitrary union of open sets is open,” gain us?
Continuity of a vector function through continuity of its components

I know that $K[X]$ is a Euclidean domain but $\mathbb{Z}[X]$ is not.

I understand this, if I consider the ideal $\langle X,2 \rangle$ which is a principal ideal of $K[X]$ but not of $\mathbb{Z}[X]$. So $\mathbb{Z}[X]$ isn’t a principal ideal domain and therefore not an Euclidean domain.

But I don’t understand this, if I consider the definition of Euclidean domains. Basically, a Euclidean domain is a ring where I can do division with remainders. For polynomial rings, the Euclidean function should be the degree of the polynomials. What’s the crucial difference between $K[X]$ and $\mathbb{Z}[X]$ with respect to this?

- What happens if we remove the requirement that $\langle R, + \rangle$ is abelian from the definition of a ring?
- Prove the direct product of nonzero complex numbers under multiplication.
- Discriminant of $x^n-1$
- Galois group of $X^4 + 4X^2 + 2$ over $\mathbb Q$.
- Determine the Galois Group of $(x^2-2)(x^2-3)(x^2-5)$
- Example of a non-splitting exact sequence $0 → M → M\oplus N → N → 0$

I already did exercises involving polynomial division in $\mathbb{Z}[X]$, so clearly I must be missing something here.

- Applications of the Jordan-Hölder Theorem.
- If $\phi(g)=g^3$ is a homomorphism and $3 \nmid |G|$, $G$ is abelian.
- Why are polynomials defined to be “formal”?
- Origin of the terminology projective module
- Order of orthogonal groups over finite field
- A module as an external direct product of the kernel and image of a function
- How to show that $(-a)(-b)=ab$ holds in a field?
- On irreducible factors of $x^{2^n}+x+1$ in $\mathbb Z_2$
- Do there exist semi-local Noetherian rings with infinite Krull dimension?
- A group of order $108$ has a proper normal subgroup of order $\geq 6$.

Try to divide something like $x+1$ by $2x+1$. If it were a Euclidean Domain, you should be able to write $x+1=q(x)(2x+1) + r(x)$ where $r(x)$ has to have degree 0. You can see why this is not possible to do by looking at the coefficient on the $x$ term, since $2$ is not invertible in $\mathbb{Z}$

In $K[x]$ the reason things go nicely is because every non-zero element in $K$ is invertible ($K$ being a field). So we can have

$$a(x)=b(x)q(x)+r(x) \qquad \text{ with } \qquad 0 \leq \text{deg}r(x) < \text{deg}b(x).$$

But one of the problems in executing this in $\mathbb{Z}[x]$ is that unless the polynomial is monic you may not have the degree inequality (**which is one the requirements of Euclidean Norm function**). For example. if you try to carry out division with $a(x)=x+1$ and $b(x)=2x$, then you cannot have the degree of the remainder less than $1$. Thereby it does not fulfill the required conditions of an Euclidean Domain.

Try perform a division with remainder between $X$ and $2$ in $\mathbb Z[X]$.

Take the two polynomial of $\mathbb{Z}[X]$, $P(X)=X^2+1$ and $Q(X)=2X$ and try to perform the division while keeping integer coefficients.

You can see the obstruction is coming from non invertible elements of the ring $\mathbb{Z}$

Suppose you have a Euclidean domain. I claim that any element $x$ with $\deg x = 0$ should be invertible. Indeed the claim of the division algorithm gives that I can write $1 = qx + r$ with $r = 0$ or $\deg r < \deg x$. The latter is impossible if the degree is $0$, so $r = 0$ and thus $qx = 1$, giving that it is a unit.

Thus the explicit obstruction is that there are integers with degree 0 that are not invertible in the polynomial ring.

In fact, let $R$ be an integral domain and consider the ring $R[x]$ with the degree function. Then if $R[x]$ were a Euclidean domain under the degree function, every $r \neq 0 \in R$ would be invertible in $R[x]$, but since $R$ is an integral domain it is easy to see this cannot happen without $r$ being invertible in $R$. So $R$ is a field.

**Hint** $\ $ An element $a\ne 0\,$ of $\,\rm\color{#c00}{minimal}\,$ Euclidean value is a unit, i.e. invertible (else $\,a\nmid b\,$ for some $\,b\,$ so $\,b\div a\,$ leaves remainder smaller than $\,a,\,$ contra minimality of $\,a).\,$ In particular, if the polynomial degree is a Euclidean function, then a nonzero element of $\,\rm\color{#c00}{degree\ 0}\,$ is a unit.

**Remark** $\ $ This is a special case of the general argument that ideals are principal in Euclidean domains, i.e. any element $\neq 0\,$ of $\,I\,$ of least Euclidean value must divide every other element, else the remainder would be a smaller element of $I.$ Above is the special case $\,I = (1).$

Similar ideas can be used to prove that certain quadratic number rings are not Euclidean, e.g. see the use of a “universal side divisor” in the proof of Lenstra linked in this answer. One can obtain a deeper understanding of Euclidean domains from the excellent exposition by Hendrik Lenstra in *Mathematical Intelligencer* 1979/1980 (Euclidean Number Fields 1,2,3)

In a PID, any irreducible element generates a maximal ideal. In $\mathbf Z[X]$, both $2$ and $X$, say are irreducible, but they’re no maximal, since:

$$\mathbf Z[X]/2\mathbf Z[X]\simeq(\mathbf Z/2\mathbf Z)[X]\quad\text{and}\mathbf Z[X]/X\mathbf Z[X]\simeq\mathbf Z,$$

which are not fields.

- When is $\left\lfloor \frac {7^n}{2^n} \right\rfloor \bmod {2^n} \ne 0\;$?
- Examples of prime ideals that are not maximal
- If $N\in \mathbb N$ is a palindrome in base $b$, can we say it is a palindrome in other bases?
- Intuitive explanation of covariant, contravariant and Lie derivatives
- Prove $f(x) = 0 $for all $x \in [0, \infty)$ when $|f'(x)| \leq |f(x)|$
- A geometric inequality, proving $8r+2R\le AM_1+BM_2+CM_3\le 6R$
- Find two closed subsets or real numbers such that $d(A,B)=0$ but $A\cap B=\varnothing$
- Give an example to show that a factor ring of a ring with divisors of 0 may be an integral domain
- If $(A-\lambda{I})$ is $\lambda$-equivalent to $(B-\lambda{I})$ then $A$ is similar to $B$
- Integral $\int_0^{\infty} \frac{\log x}{\cosh^2x} \ \mathrm{d}x = \log\frac {\pi}4- \gamma$
- If $x \equiv 1 \pmod 3$ and $x \equiv 0 \pmod 2$, what is $x \pmod 6$?
- Find the latus rectum of the Parabola
- product of harmonic functions
- Prove that if $7^n-3^n$ is divisible by $n>1$, then $n$ must be even.
- How do you explain the concept of logarithm to a five year old?