# Nicolas Lykke Iversen

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# 177 Questions

 10 If $A \in M_{n,n}(\mathbb F)$ is invertible then $A = UPB$, $U$ is unipotent upper triangular, $B$ is upper triangular and $P$ is a permutation. 6 Let $R = \mathbb Z[i]$. Show $I \cap \mathbb Z$ is an ideal in $\mathbb Z$, for all $a \in I \cap \mathbb Z$, $10 \mid a^2 = N(a)$. 4 Prove that $\int_{0}^1 u^{\alpha_1-1} (1-u)^{\alpha_2-1} \, {\rm d}u =\frac{\Gamma(\alpha_1)\Gamma(\alpha_2)} {\Gamma(\alpha_1+\alpha_2)}$ 4 Why is an extension $\bar \phi : F(x) \rightarrow F(a)$ an isomorphism if $\phi : F[x] \to F(a)$ is injective? 4 Why is the Legendre symbol not defined for $p = 2$ (even prime)?

# 1,176 Reputation

 +10 Prove that $\int_{0}^1 u^{\alpha_1-1} (1-u)^{\alpha_2-1} \, {\rm d}u =\frac{\Gamma(\alpha_1)\Gamma(\alpha_2)} {\Gamma(\alpha_1+\alpha_2)}$ +10 Prove a residue matrix $A$ (with coefficients in $\mathbb Z_n)$ has an inverse if and only if $\gcd(\det A,n) = 1$ +5 Dual problem of a maximization primal problem $P$? +5 Prove the improper integral of the Gamma function $\Gamma(t)$ converges for $z \in \mathbb C$ with $Re(z) > 0$:

# 10 Answers

 6 $F$ is a field $\implies$ $1_F \neq 0_F$ 4 Show $x^2 \equiv 94 \pmod{195}$ if and only if $x^2 \equiv 1 \pmod 3$, $x^2 \equiv 4 \pmod 5$ , $x^2 \equiv 3 \pmod{13}$ 3 Let $G$ be a graph. $G$ is graceful $\Rightarrow G$ is connected? 2 Let $R = \mathbb F_3[X]/\langle X^3 + X^2 + 1\rangle$ and $\alpha = [X] \in R$. Show $R^*$ is not cyclic. 2 Show there exists precisely one number $\alpha$ where $0 \le \alpha < 210$ that solves the system of congruence equations.

# 72 Tags

 8 abstract-algebra × 43 1 group-theory × 5 8 elementary-number-theory × 28 1 groebner-basis × 3 6 ring-theory × 4 0 linear-algebra × 35 3 graph-theory × 7 0 real-analysis × 25 2 number-theory × 10 0 sequences-and-series × 23

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