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Prove that $n ∣ ϕ (a^{n} - 1)$ in "Topics in Algebra 2nd Edition" by I. N. Herstein. Any natural solution that uses $Aut (G)$

redupticslaz 2022-04-27

I have to prove that if P is a R-module , P is projective right there is a family ${x_{i}}$ in P and morphisms $f_{i} : P \Rightarrow R$ such that for all $x \in P$
$x = \sum_{i \in I} f_{i} (x) x_{i}$
where for each $x \in P, f_{i} (x) = 0$ for almost all $i \in I$ .

slanglyn3u2 2022-04-25

Socle of socle of module is the socle of the module
$S o c (S o c (M)) = S o c (M)$

Aleena Kaiser 2022-04-25

Show that $gcd (a, b) = | a | \Leftrightarrow a ∣ b$ ?

haguemarineo6h 2022-04-25

Polynomial satisfying $p (x) = 3^{x}$ for $x \in N$

Kymani Shepherd 2022-04-24

Number of Elements of order p in $S_{p}$
An exercise from Herstein asks to prove that the number of elements of order p, p a ' in $S_{p}$ , is $(p - 1)! + 1$ . I would like somebody to help me out on this, and also I would like to know whether we can prove Wilson's theorem which says $(p - 1)! \equiv - 1$ (mod p) using this result

Jakayla Benton 2022-04-24

Show that $\sqrt[3]{π}$ is transcendental

et3atissb 2022-04-23

Is k[[x]] ever a finitely generated k[x](x) module?
For k a field, the localization $k {[x]}_{(x)}$ naturally includes into k[[x]]. I can prove that if $k = C,$ , then this inclusion is not surjective, and k[[x]] is not even finitely generated over $k {[x]}_{(x)}$ , because $C {[x]}_{(x)}$ corresponds to rational functions holomorphic at 0, and $C [[x]]$ corresponds to germs of all functions holomorphic at 0, and so with some complex analysis you can see the finite generation is impossible. But over an arbitrary field, is this claim still true?

windpipe33u 2022-04-23

Show that the Lorentz boosts,
$(\begin{matrix} γ & - β γ \\ - β γ & γ \end{matrix})$
form a one-parameter Lie group.

coraletsmmh 2022-04-23

Product of two cyclic groups is cyclic iff their orders are co-'
Say you have two groups $G = ⟨ g ⟩$ with order n and $H = ⟨ h ⟩$ with order m. Then the product $G \times H$ is a cyclic group if and only if $gcd (n, m) = 1$

Zack Wise 2022-04-22

Notation of homeomorphism from B(H) to B(K), corresponding to unitary transformation of Hilbert spaces
Let U be a unitary transformation from Hilbert space H to Hilbert space K.
How do you call a *-homomorphism f from B(H) to B(K), defined by $f (a) = U a U^{- 1}$ ?
I'm interested both in a symbol, which can be used in formulas, and a name for it, which can be used in texts or in speech.

Saige Shannon 2022-04-22

How does cancellation work in polynomial quotient rings?
$\frac{Z [x]}{(2 x - 6, 6 x - 15)}$
Can I just automatically say that
$\frac{Z [x]}{(2 x - 6, 6 x - 15)} \tilde{=} \frac{Z [x]}{(x - 3, 6 x - 15)}$
by just dividng the first polynomial by 2?

Dereon Guzman 2022-04-21

If the cardinality of a ring R is greater than 1, then $1 \neq 0$ .

devojciq5o 2022-04-20

Solving (quadratic) equations of iterated functions, such as
$f (f (x)) = f (x) + x$

Jaylyn Villarreal 2022-04-17

We want to prove that $(Z_{7}, \oplus)$ is group. I have difficulty proving associativity axiom. The solution reads
Let $a \in Z_{7}, b \in Z_{7}$ and $c \in Z_{7}$ . By Theorem 3.4.10 we only need to show
$(a + (b + c)) b mod 7 = ((a + b) + c) b mod 7 .$
This holds since $a + (b + c) = (a + b) + c$ for all integers a, b, and c by the associative property of the integers. Hence $\oplus$ is associative.
Therorem 3.4.10. Let a and b be integers, and let m be a natural number. Then
$(a + b) mod m = ((a mod m) + (b mod m)) mod m$

Kendal Day 2022-04-16

Multiples of 4 as sum or difference of 2 squares
Is it true that for any $n \in N$ we can have $4 n = x^{2} + y^{2}$ or $4 n = x^{2} - y^{2}$ , for $x, y \in N \cup (0)$ ?
I was just working out a proof and this turns out to be true from $n = 1 to n = 20$ . After that I didn't try, but I would like to see if a counterexample exists for a greater value of n.

gabolzm6d 2022-04-16

What are all the homomorphisms between the rings $Z_{18} and Z_{15}$ ?
Any homomorphism $ψ$ between the rings $Z_{18} and Z_{15}$ is completely defined by $ψ (1)$ . So from
$0 = φ (0) = φ (18) = φ (18 \cdot 1) = 18 \cdot φ (1) = 15 \cdot φ (1) + 3 \cdot φ (1) = 3 \cdot φ (1)$
We get that $ψ (1)$ is either 5 or 10. But how can I prove or disprove that these two are valid homomorphisms?

Essence Ingram 2022-04-16

We were learning about centralizers in my abstract algebra class, and I had a thought about centralizers.
Is every subgroup of a group G the centralizer of some element? And if not, how many of the subgroups are?

Papierskiix5n 2022-04-15

Let A be a discrete valuation ring, and let $a \in A$ be a non-zero element. Compute the integral closure of $B : = A [X, \sqrt{a X}]$

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