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Abstract algebra

Finding a subgroup of $_{{8}}$ isomorphic to $Z_{4} \times Z_{4}$
What might be a more clever way to go about solving this?

Mary Reyes 2022-01-12

Let A be a DVR in its field of fractions F, and $F^{'} \subset F$ a subfield. Then is it true that $A \cap F^{'}$ is a DVR in $F^{'} ?$ I can see that it is a valuation ring of $F^{'}$ , but how do I show, for example, that $A \cap F^{'}$ is Noetherian?

Ernest Ryland 2022-01-12

Is it true that for a Group G with Normal Group $N : \frac{G}{N} = \frac{G N}{N} ?$
I think the statement is correct. But why do we have to write: $[G, G] \frac{N}{N}$ here instead of just $\frac{G, G}{N} ?$

Kathy Williams 2022-01-12

Find subgroup $⟨ a, b ⟩$ of $Z_{20}^{⋆}$ which is not cyclic.

kihanja20 2022-01-12

Find the order $5 + H$

Chris Cruz 2022-01-12

How do I find $a, b \in Z$ s.t.
${a c - b d + i (a d + b c) ∣ c, d \in Z}$
have real and imaginary parts both even or both odd?

agreseza 2022-01-12

Let $H \leq G$
Prove
$x^{- 1} y^{- 1} x y \in H \forall x$
$y \in G \Leftrightarrow H ⊴ G$
and
$\frac{G}{H}$ is abelian.

Carla Murphy 2022-01-12

Let
$ϕ \in E n d (G)$ s.t.
$\exists n \geq 0, k e r (ϕ^{n}) = G$
If $k e r ϕ$ or $[G : I n_{ϕ}]$ is finite, then G is finite.

Madeline Lott 2022-01-12

Let H be a subgroup of G and $x, y \in G$ . Show that
$x (H y) = (x H) y$

piarepm 2022-01-12

Classify via fundamental theorem of finitely generated abelian groups.
$\frac{Z \times Z}{⟨ \begin{array}{cc} 3 & 1 \\ 8 & 2 \end{array} ⟩}$

Josh Sizemore 2022-01-02

Can you find a domain where $a x + b y = 1$ has a solution for all a and b relatively prime, but which is not a PID?

Teddy Dillard 2022-01-02

Is there an algorithm for working out the best way (i.e. fewest multiplications) of calculating $A^{n}$ in a structure where multiplication is associative?

chezmarylou1i 2022-01-01

Prove $x^{6} - 6 x^{4} + 12 x^{2} - 11$ is irreducible over $Q$

petrusrexcs 2021-12-30

Are there any interesting and natural examples of semigroups that are not monoids (that is, they don't have an identity element)?

Sapewa 2021-12-30

Let F be a field. Suppose that a polynomial $p (x) = a_{0} + a_{1} x + \dots + a_{n} x^{n}$ is reducible in $F [x]$ . Prove that the polynomial $q (x) == x^{n} p (\frac{1}{x}) = a_{n} + a_{n - 1} x + \dots a_{0} x^{n}$ is also reducible.

hunterofdeath63 2021-12-12

We're trying to understand the field $\frac{Z_{2} [x]}{⟨ x^{3} + x^{2} + 1 ⟩}$ , and in particular why it has eight elements. I know that $x^{3} + x^{2} + 1$ is irreducible in $Z_{2}$ and so factor is going to be a field.
The issue I'm having is that the notes claim that $(x + I)$ has an inverse in the field, where $I = ⟨ x^{3} + x^{2} + 1 ⟩$ . They write
$(x + I) (x^{2} + x + I) = x^{3} + x^{2} + I = 1 + I$
I don't understand the last line, I know that multiplication of cosets is defined that way, I just don't see why we get 1 at the end. I think it's possible I didn't fully understand cosets and ideals but I can't fully articulate what the issue is. Any help towards filling in the gaps of my understanding would be greatly appreciated.

hexacordoK 2021-10-09

If a, b are elements of a ring and m, n ∈ Z, show that (na) (mb) = (mn) (ab)

Suman Cole 2021-09-19

Explain why (a) Z_9 is not isomorphic to Z_3 × Z_3.
(b) Z_9 × Z_9 is not isomorphic to Z_9 × Z_3 × Z_3.

ddaeeric 2021-06-23

Let $n \geq 8$ . Give a $b i g - Θ$ estimate for the number of circuits of length 8 in Kn. $Θ (n) Θ (n 8) Θ (8) Θ (8 n) Θ (8 n)$

permaneceerc 2021-06-19

The front wheel of a tricycle has a circumference of 63 inches, and the back wheels have a circumference of36 inches. If points P and Q are both touching the sidewalk when Felicity starts to ride, how far will she have ridden when P and Q first touch the sidewalk at the same time again?

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