Let $f=X^n-1$ and $g=X^m-1$ be two polynomials. Show that: $$\left(f,g\right)=X^{\left(n,m\right)}-1,$$ where $\left(a,b\right)=$ greatest common divisor of $a$ and $b$.
2481
For all $a, m, n \in \mathbb{Z}^+$,   $$\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$$
45033
I don't really know how to prove this, i suppose it must have something to do with the definition of greatest common divisor and Bézout's identity.  I looked at some other questions here and thought that i had to get to gcd(ma+nb,mn)=1 and that gcd(a,n)gcd(b,m)=1 too, but i'm not sure.
40355
Why is $\gcd(X^n - 1, nX^{n-1}) = 1$ in a field $K$ of characteristic $0$ or characteristic $p \nmid n$.  Most book on abstract algebra don't prove explicitly this, but I wonder what technique I should use to prove this to be formal. Perhaps induction?
63271
Here is another problem of the book Fields and Galois Theory by Patrick Morandi, page 37.     Let $f(x)$ be an irreducible polynomial over $F$ of degree $n$, and let $K$ be a field extension of $F$ with $[K:F]=m$. If $\gcd(n,m)=1$, show that $f$ is irreducible over $K$.   I'm preparing for my midterm exam so I'm trying to solve as many as this book problems. Thanks for your helps.
109287
For a polynomial $f=X^n+a_1X^{n-1}+\ldots+a_n \in \mathbb{Q}[X]$ we define $\varphi(f):=a_1 \in \mathbb{Q}$. Now I want to show that for the $n$th cyclotomic polynomial $\Phi_n$ it holds that $$\varphi(\Phi_n)=-\mu(n)$$ where $\mu(n)$ is the Möbius function. What I know is that $\displaystyle\Phi_n=\prod_{d|n} (X^{\frac{n}{d}}-1)^{\mu(d)}$.
106515
Let $k$ be any field, and let $f,g\in k[x,y]$ be two irreducible polynomials such that $g$ is not divisible by $f$. Prove that $V(f,g)\subseteq A_k^2$ is finite.
123064
I am working on a problem from Ideas, Varieties, and Algorithms:     Show that if $f \in \mathbb{C} \left[x_1, \dots, x_n \right]$ vanishes at every point of $\mathbb{Z}^n$, then $f$ is the zero polynomial.   My attempt so far:  This proof is by induction on $n$. Let $f \in \mathbb{C}\left[x\right]$ vanish at every point in $\mathbb{Z}$. Then, it must be identically the zero polynomial, as a polynomial of degree $d$ has at most $d$ roots. Hence, $f$ vanishes at every point at $\mathbb{C}$. Now, assume that $g \in \mathbb{C} \left[x_1, \dots, x_{n-1} \right]$ vanishes at every point in $\mathbb{Z}$. We may collect terms and rewrite $f$ as  $$ f(x_1, \dots, x_n) = \sum_{i=0}^m x^i_n g(x_1, \dots, x_{n-1}), $$ for some functions $g_1,\dots,g_n\in \Bbb C[x_1,\dots,x_{n-1}]$.  How do I proceed from here? I was trying to adapt an earlier proof with no success.
105683
We know that if $F$ is a field and $f(X)$ a non-zero polynomial in $F[X]$, then for every polynomial $g(X)$ we can find $q(X),r(X)$ such that $$g(X)=f(X)\cdot q(X)+r(X)$$ with $r(X)$ the zero polynomial or $\deg r(X)&lt;\deg f(X)$. My question is: the same algorithm holds for many variables? So can i write $$g(X_1,\ldots,X_n)=f(X_1,\ldots,X_n)\cdot q(X_1,\ldots,X_n)+r(X_1,\ldots,X_n)$$
46683
Find all polynomials $F(x)={a_n}{x^n}+\cdots+{a_1}x+a_0$ satisfying   $a_n \neq0$; $(a_0, a_1, a_2, \ldots ,a_n)$ is a permutation of $(0, 1, 2 ... n)$; all zeros of $F(x)$ are rational.
63983
I have this problem      Let $L/K$ be a finite field extension and $|L:K|=m$. Let $f(x)$ be an irreducible polynomial of degree $n$ in $K[x]$ and $\gcd(m,n)=1$. Prove that $f$ is irreducible over $L$.   I'm just a beginner so I'm really confused with the $\gcd(m,n)=1$, I don't know how to use it. So help me with this problem. Thank you
73433
Suppose $z_1, .....z_{m+1}$ are distinct elements of $F$ and $w_1,....,w_{m+1} \in F$. Prove that there exists a unique polynomial $p \in P_m(F)$ such that $p(z_j) = w_j$ for j=1,...m+1.  Any ideas on where to start/go with this one. I'm having quite some trouble with it.
17420
Let $F$ be a field and $x$ transcendental over $F$. Let $y=f(x)/g(x)$ be a nonconstant rational function with relatively prime polynomials $f,g\in F[x]$. Prove $[F(x):F(y)]=\max(\deg f,\deg g)$.  My attempt: By replacing $y$ with $\frac{1}{y}$ if necessary we assume $\deg g\geq\deg f$. By the Euclid algorithm we assume $\deg g&gt;\deg f$. Then $\deg g=\max(\deg f,\deg g)$, let $n=\deg g$. Let $R(t)$ be this polynomial $$R(t)=yg(t)-f(t)=\frac{f(x)}{g(x)}g(t)-f(t)$$ Then $\deg R=\deg g=n$ because $n=\deg g&gt;\deg f$. The coefficients of $R(t)$ lie in $F(y)$ and hence $R(t)\in F(y)[t]$. Clearly $R(x)=0$.  Now I only have to prove $R(t)$ is actually the minimal polynomial of $x$ over $F(y)$. I believe the next step is to obtain a contradiction that $R(t)$ being reducible can lead to $f,g$ being not coprime, but I can't figure out how to do that.  P.S. A hint is preferred over a full answer.
37895
Let $E/F $ be a field extension and $f,g$ $\in$ $F[x]$ (the polynomial ring with coefficients in $F$ ). Let's denote with $\gcd_F(f,g)$ the greatest common divisor of $f$ and $g$ in $F[x]$. Is it true that $\gcd_F(f,g)$ $=$ $\gcd_E(f,g)$ ?
94023
I need your help with this: Let $f$, $p$, and $g$ be a polynomials in $F[x]$, assume that $f$, $g$, and $p$ are nononzero.  If $\gcd(f,p)=1$ and $fg$ is divisible by $p$ with no remainder, I need to prove that $g$ is  divisible by $p$ with no remainder  Thanks.
123202
This is problem 31b) from Dummit and Foote chapter 14.2. I am looking for a hint on how to attack the problem, since I have been thinking about it for a couple of hours but I don't even know where to start. The problem says:  Let $K$ be a finite extension of $F$ of degree $n$. Let $\alpha$ be an element of $K$. Prove that the minimal polynomial for $\alpha$ over $F$ is the same as the minimal polynomial for the linear transformation $T_{\alpha}$. In this problem, $T_{\alpha}$ is an $F$-linear transformation of $K$ that arises from $\alpha$ acting by left multiplication on $K$.  I appreciate any helpful suggestions on how to start attacking the problem or any possible hint. Thanks!
116448
Given a polynomial $f(x)$ of n degree such that $$f(x)+f\left(\frac{1}{x}\right)=f(x)\cdot f\left(\frac{1}{x}\right)$$ Find the polynomial  I've tried considering $f(x)=\sum \limits_{i=1}^{n} a_{i}x^{i}$ and after a number of steps I finally arrived at the answer.However I am not convinced.Is there a better way to solve this functional equation. Thanks in advanced!
29217
How do we find the GCD $G$ of two polynomials, $P_1$ and $P_2$ in a given ring (for example $\mathbf{F}_5[x]$)? Then how do we find polynomials $a,b\in \mathbf{F}_5[x]$ so that $P_1a+ P_2b=G$? An example would be great.