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Number of elements in $S_{13}$ conjugate to both $(12) (34)$ and $(123) (45)$ then is the answer 0? Because $(12) (34)$ and $(123) (45)$ have not got the same cycle structure.

sunshine022uv 2022-01-14

Find a polynomial $P_{n} \in C [X]$ such as $S_{n} = k e r P_{n} (D)$
With $S_{n} (n \geq 1)$ all of the functions such as $y^{(n)} = y$ (where $y^{(n)}$ is the nth derivative of y)
D the endomorphism sending the functions $C^{\infty}$ on their derivatives $D : y \to y^{'}$
I already proved that D is an endomorphism and $S_{n}$ a vector space containing the functions $k e^{x}$ .
But my probleme is that i dont really understand the meaning of $P_{n} (D)$ , and so dont see what $S_{n} = k e r P_{n} (D)$ could be. It probably have to do with eigenvalues, eigenvectors...But really I dont see anything.

Pamela Meyer 2022-01-13

If $\frac{K}{F}$ is Galois, and the coefficients of the monic $f \in K [x]$ generates K, then $\prod_{σ \in G a l (\frac{K}{F})} σ f \in F [x]$
How does one see that $g \in F [x]$ ?

Mabel Breault 2022-01-13

This is a well known lemma that consistently appears in textbooks, either as a statement without proof, or as an exercise.
If $0 \underset{\to}{i d} A \underset{\to}{f} B \underset{\to}{g} C \underset{\to}{h} 0$ is a short exact sequence of finitely generated abelian groups, then
$rank B = rank A + rank C$
I've been trying to prove this unsuccessfully.
What do we know? f is injective, g is surjective,
$I m f = k e r g, I m g = k e r h, C \tilde{=} \frac{B}{A}$
So I start with a maximally linearly independent subset ${a_{α}}$ of A such that the sum (with only finite non-zero entries)
$\sum n_{α} a_{α} = 0$
for $n_{α} \in Z$ , implies that $n_{α} = 0$
Where to go from here is a puzzle?

socorraob 2022-01-13

Let k be a field and R be the exterior ring over $k^{d}$ , that is, k-algebra genarated by elements
$x_{1}, \dots . x_{d},$
where
$x_{i} x_{j} = - x_{j} x_{i}$ ?
Is R Artinian?

William Collins 2022-01-13

What reduced quadratic equation has solutions $\sqrt{2}$ and $\sqrt{3}$ ?

widdonod1t 2022-01-13

Factor $x^{4} - 2$ into irreducible polynomials.

James Dale 2022-01-13

Abelian group G such that the infinite-order elements form a subgroup with the identity.
If ${g \in G ∣ g = e or g has infinite order}$ is a subgroup of G, what can we say about the order of the elements of G?

Krzychau1 2022-01-13

What is the set $\frac{Z}{2 Z} ?$

Sallie Banks 2022-01-13

I've been trying to find a rig-like structure (a set R with a monoid structure $(R, \cdot, 1)$ and a (commutative) monoid structure $(R, +, 0)$ such that the multiplication distributes over addition) in which 0 is not an absorber with respect to multiplication, i.e. $0 \cdot r \neq 0$ for some $r \in R$ .
It should be possible to construct such a structure since you have to require absorption as a separate axiom in a rig.
So far I haven't been successful as I don't have any experience with rigs but maybe someone else has a neat idea?

Irvin Dukes 2022-01-13

When does the produt of two polynomials $= x^{k}$ ?
Suppose f and g are are two polynomials with complex coefficents (i.e $f, g \in C [x]$ ). Let m be the order of f and let n be the order of g.
Are there some general conditions where
$f g = α x^{n + m}$
for some non-zero $α \in C$

Russell Gillen 2022-01-13

Does there exists a Ring with unity other than $Z_{4}$ such that for all non-zero non-unit elements of ring, $2 x = 0 and x \cdot x = 0$
Let R be a ring with unity and $x + x = 0, x \cdot x = 0$ for all non-zero non unit element x of R.
$Z_{4}$ is one of the example of that ring. Does there exists any other ring of this type ? If no then how can we prove that.

Zerrilloh6 2022-01-13

Given a subspace V of a metric Lie algebra, does $[[V, V], V] \subset V$ imply $[V, V] \subset V ?$

sunshine022uv 2022-01-13

Why must a be in one of these two cosets: $H, H a ?$

crealolobk 2022-01-13

Aluffis

kihanja20 2022-01-13

Prove that:
Every $Q$ - vector space $V \neq 0$ is not free over subring $Z \subset Q$

osula9a 2022-01-13

Prove $k e r ϕ = ⟨ 1 + \sqrt{- 5}, 2 ⟩$

Mary Reyes 2022-01-13

Let $α = (1 7 3) (5 4 2 9) and β = (2 3) (7 4) (5 1 8)$ be elements of $S_{10} .$ Then $α^{2} β^{- 1}$ is:
a) $(1 2 7 9 4) (3 8 5)$
b) $(1 2 7 4 9) (3 5 8)$
c) $(1 2 7 9 4) (3 5 8)$
d) $(1 2 9 7 4) (3 5 8)$

Daniell Phillips 2022-01-13

Why does $K ⇝ K (X)$ preserve the degree of field extensions?

Algotssleeddynf 2022-01-12

An automorphism of a group G is an isomorphism from G to itself. Denote by Aut G the set of all automorphisms of G.
a) Prove that Aut G is a group with respect to the operation of composition
b) Give an example of an abelian G such that Aut G is not abelian

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