Found problems: 9
Prove that for any polynomial $P$ with real coefficients, and for any positive integer $n$, there exists a polynomial $Q$ with real coefficients such that $P(x)^2 +Q(x)^2$ is divisible by $(1+x^2)^n$.
Let $K$ be a finite field with $q$ elements, $q \ge 3.$ We denote by $M$ the set of polynomials in $K[X]$ of degree $q-2$ whose coefficients are nonzero and pairwise distinct. Find the number of polynomials in $M$ that have $q-2$ distinct roots in $K.$
[i]Marian Andronache[/i]
Let $K$ be a finite field, $n \ge 2$ an integer, $f \in K[X]$ an irreducible polynomial of degree $n,$ and $g$ the product of all the nonconstant polynomials in $K[X]$ of degree at most $n-1.$ Prove that $f$ divides $g-1.$
Show that the equation $ (x+y)^{-1}=x^{-1}+y^{-1} $ has a solution in the field of integers modulo $ p $ if and only if $ p $ is a prime congruent to $ 1 $ modulo $ 3. $
[i]Mihai Piticari[/i]
Show that the set of all elements minus $ 0 $ of a finite division ring that has at least $ 4 $ elements can be partitioned into two nonempty sets $ A,B $ having the property that
$$ \sum_{x\in A} x=\prod_{y\in B} y. $$
Prove that for any polynomial $P$ with real coefficients, and for any positive integer $n$, there exists a polynomial $Q$ with real coefficients such that $P(x)^2 +Q(x)^2$ is divisible by $(1+x^2)^n$.
Let be two nontrivial rings linked by an application ($ K\stackrel{\vartheta }{\mapsto } L $) having the following properties:
$ \text{(i)}\quad x,y\in K\implies \vartheta (x+y) = \vartheta (x) +\vartheta (y) $
$ \text{(ii)}\quad \vartheta (1)=1 $
$ \text{(iii)}\quad \vartheta \left( x^3\right) =\vartheta^3 (x) $
[b]a)[/b] Show that if $ \text{char} (L)\ge 4, $ and $ K,L $ are fields, then $ \vartheta $ is an homomorphism.
[b]b)[/b] Prove that if $ K $ is a noncommutative division ring, then it’s possible that $ \vartheta $ is not an homomorphism.
Let $(K,+,\cdot)$ be a field with the property $-x=x^{-1},\forall x\in K,x\ne0$. Prove that:
$$(K,+,\cdot)\simeq(\mathbb Z_2,+,\cdot)$$
[i]Proposed by Ovidiu Pop[/i]
Let $ u,v $ be two endomorphisms of a finite vectorial space that verify the relation $ uv-vu=u. $
Calculate $ u^kv-vu^k $ and show that u is nilpotent.