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comment Show that $ζ$ is a Quadratic Integer in $Q[\sqrt{−3}]$
have you tried simply computing $(\frac{a+b\sqrt{-2}}2)^3$?
comment Solving the functional equation $f(x^2+f(y))=(f(x))^2+y$
How can negative identity be a solution when $f$ is nondecreasing per $(3)$?
comment Integral of absolute value of polynomial?
If $\alpha,\beta$ are the two roots of $b$, then $$\int_{-m}^m|b(x)|\,\mathrm dx = \int_{-m}^mb(x)\,\mathrm dx-2\int_{\alpha}^\beta b(x)\,\mathrm dx=a(m)-a(-m)-2a(\beta)+2a(\alpha) $$ So the question is whether $a(\alpha)-a(\beta)$ can be expressed by radicals.
comment Sum of two normal numbers need not be a normal one
Actually, if one doesn't know what normal means one would not even guess that $d_i(\omega)$ denotes the $i$th binary digit of $\omega$ ...
comment Proving that $F_{kn}$ is a multiple of $F_n$ by induction on $n$ (Fibonacci numbers)
Yeah, make your choice so that the proof works ...
comment Assumptions, Axioms and Premises
Maybe the attempt to make a somewhat formal distiction between these is as moot as for a distinction between lemma, proposition, theorem, corollary.
comment Proof of infinite monkey theorem.
@AdityaAgarwal it is a theorem about a probability hence probabilty theory is the tool to use. Also, Borel-Cantelli is more or less equivalent to the infinite monkey theorem. The idea is to compare convergence of $1-\prod(1-p)$ with that of $\sum p$
comment Is there a symbol for “given” in mathematics?
Actually, I wouldn't use "given" in this context for the natural language version.
comment Algebraic groups?
@MilesUbiquity with the operation $a\circ b=a+b-1$, the real number from a group with neutral element $1$. In fact, $x\mapsto x-1$ is a group isomorphism $(\mathbb R,\circ)\to(\mathbb R,+)$.
comment If $B$ is a maximal linearly independent set in $V$ then $B$ is a basis for $V$
@zed111 if $\alpha_{n+1}=0$ then $\alpha_{n+1}x$ can be ignored and the dependence relation is really among the $b_i$ alone, but $B$ was assumed to be linearly independent.
comment Uniform Polyhedron with 500 congruent right kite faces!
I suppose the very acute angles are all at the north and south pole and the length $a$ wiggle about the equator?
comment Prove that every nearly euclidean domain is a principal ideal domain.
"I'm trying the prove that every principal ideal domain is a nearly euclidean domain"?
comment Prove that every nearly euclidean domain is a principal ideal domain.
It's not a typo. If $a \ne 0$, you write $b=qa+r$ in division with remainder, so the $1$ must be with $b$.
comment $R^2$ a subspace of $R^3$
You have a straightforward embedding $\mathbb R^2\to\mathbb R^3$, $(x,y)\mapsto (x,y,0)$. But allows a straightforward embedding is not the same as is a subset. Especially, nothing prevents your neighbour from considering $(x,y)\mapsto (0,x,y)$ straightforwrd.
comment Galois group of $x^4-5$ over $\mathbb{Q}$.
The keyword $2$-Sylow group might be worth mentioning as well ...
comment Galois group of $x^4-5$ over $\mathbb{Q}$.
Actually, you could refine the tower to $\mathbb Q\subset \mathbb Q[\sqrt 5]\subset \mathbb Q[\sqrt[4]5]\subset K$.
comment Let $\varphi$ be a group homomorphism from $G$ to $H$. Let $\varphi(G)$ be its image and $K$ its kernel. Show that $K$ is a normal divisor of $G$.
Look up and apply the definition, that's all there is to it.
comment $A,B \in {M_n}$ are normal.why the null space of $A$ is orthogonal to the range of $A$?
And what does $B$ have to do with this? - And could it be one of th $A$ is intended to be $A^*$?
comment What is the example of a not almost convergent sequence but whose Cesàro means converge?
Your example of the second paragraph is not Cesàro because $\frac1nS_n\approx \pm \frac 12$. Maybe $x_i=(-1)^i\sqrt i$?
comment Two pawns walking in a complete graph
From a practical standpoint, $A^*$ might be a good idea