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Prove that $a\equiv b\pmod p\Rightarrow a^p \equiv b^p \pmod {p^2}$

score 4.8k · Answer 1 · 2026-04-16 16:01:45

4.8k

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Prove that $a\equiv b\pmod p\Rightarrow a^p \equiv b^p \pmod {p^2}$

Published on 16 Apr 2026 - 16:01

score 2.6k · Answer 2 · 2026-04-17 00:37:16

2.6k

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$f(x) \text{ irreducible in } R[x] \iff f(ax+b) \text{ irreducible in } R[x] $

Published on 17 Apr 2026 - 0:37

score 30.8k · Answer 3 · 2026-04-09 02:10:17

30.8k

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Proof verification: the product of two continuous functions is continuous

Published on 09 Apr 2026 - 2:10

score 313 · Answer 4 · 2026-04-17 14:42:22

313

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Find all holomorphic functions $f$ in $D(0;1)$ such that $f\left(\frac{1}{n}\right)=n^2f\left(\frac{1}{n}\right)^3$

Published on 17 Apr 2026 - 14:42

score 2.8k · Answer 5 · 2026-04-11 15:19:48

2.8k

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If $H$ and $K$ are abelian normal subgroups of a group that intersect trivially, then prove $HK$ is abelian.

Published on 11 Apr 2026 - 15:19

score 89 · Answer 6 · 2026-04-10 23:46:16

89

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$3\times 3$ Matrix Completion

Published on 10 Apr 2026 - 23:46

score 6.4k · Answer 7 · 2026-04-12 02:55:28

6.4k

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Associativity of symmetric set difference

Published on 12 Apr 2026 - 2:55

score 82 · Answer 8 · 2026-04-14 22:42:54

82

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Find the $\sup$ of the set.

Published on 14 Apr 2026 - 22:42

score 341 · Answer 9 · 2026-04-17 14:07:47

341

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If $f^{-1}(f(E)) = E$ for all $E \subseteq X$ then $f(A-B) = f(A) - f(B)$ for all $A,B\subseteq X$

Published on 17 Apr 2026 - 14:07

score 3.4k · Answer 10 · 2026-04-17 04:40:37

3.4k

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Show that a finite field cannot be ordered

Published on 17 Apr 2026 - 4:40

score 21k · Answer 11 · 2026-04-15 10:53:48

21k

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Show that if $p$ is prime then $\Bbb Z_p$ is a field

Published on 15 Apr 2026 - 10:53

score 425 · Answer 12 · 2026-04-12 07:17:28

425

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Showing that the union of transitive relations need not be transitive.

Published on 12 Apr 2026 - 7:17

score 10.4k · Answer 13 · 2026-04-14 20:16:50

10.4k

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Prove or disprove: if $R$ and $S$ are two equivalence relations on a set $A$ then $R\cup S$ is also an equivalence relation.

Published on 14 Apr 2026 - 20:16

score 150 · Answer 14 · 2026-03-25 09:51:16

150

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Show an $n\times n$ matrix $A$ is sim. to the companion matrix for $p_A(t)\iff \exists$ a vector $x$ such that $x,Ax,\ldots,A^{n-1}x$ is a basis

Published on 25 Mar 2026 - 9:51

score 628 · Answer 15 · 2026-04-15 09:04:05

628

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Erratum for Billingsley’s $\textit{Probability and Measure}$, Problem 32.13

Published on 15 Apr 2026 - 9:04

List Question

Prove that $a\equiv b\pmod p\Rightarrow a^p \equiv b^p \pmod {p^2}$

$f(x) \text{ irreducible in } R[x] \iff f(ax+b) \text{ irreducible in } R[x] $

Proof verification: the product of two continuous functions is continuous

Find all holomorphic functions $f$ in $D(0;1)$ such that $f\left(\frac{1}{n}\right)=n^2f\left(\frac{1}{n}\right)^3$

If $H$ and $K$ are abelian normal subgroups of a group that intersect trivially, then prove $HK$ is abelian.

$3\times 3$ Matrix Completion

Associativity of symmetric set difference

Find the $\sup$ of the set.

If $f^{-1}(f(E)) = E$ for all $E \subseteq X$ then $f(A-B) = f(A) - f(B)$ for all $A,B\subseteq X$

Show that a finite field cannot be ordered

Show that if $p$ is prime then $\Bbb Z_p$ is a field

Showing that the union of transitive relations need not be transitive.

Prove or disprove: if $R$ and $S$ are two equivalence relations on a set $A$ then $R\cup S$ is also an equivalence relation.

Show an $n\times n$ matrix $A$ is sim. to the companion matrix for $p_A(t)\iff \exists$ a vector $x$ such that $x,Ax,\ldots,A^{n-1}x$ is a basis

Erratum for Billingsley’s $\textit{Probability and Measure}$, Problem 32.13

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