Abstract Algebra

Given two polynomials in $Z_2[X]$, add them and simplify the result.

Multiply the polynomials $X^3 + X$ and $X^2 + X + 1$ in $Z_2[X]$ and simplify the result.

Add $3X^3 + 2X^2 + 1$ and $3X^3 + X^2 + 2X + 2$ in $Z_4[X]$ and simplify the result modulo 4.

Multiply the polynomials $3X^2 + 2X + 3$ and $2X + 3$ in $Z_4[X]$ and simplify the result modulo 4.

Let the set $\mathbb{Z}[x]$ be the ring of polynomials over integers. Then, the additive group $\mathbb{Z}[x]$ is either isomorphic to the multiplicative group of positive rational numbers, or isomorphic to the group of rational numbers $\mathbb{Q}$ under addition, or countable, or uncountable.

Given: $\phi$ is a homomorphism from $G$ to $H$, $H$ is an abelian group, and the kernel of $\phi$ is a subset of $N$, where $N$ is a subgroup of $G$. Prove that $N$ is a normal subgroup of $G$.

Given a finite Abelian group G of order 16, enumerate all possible isomorphic groups by considering the factorization of order 16.

Using the Fundamental Theorem of Finite Abelian Groups, list all the groups of order 540 by considering its prime factorization.

How many finite Abelian groups are there of order 10,000 (10,000 being $2^4 \times 5^4$)?

How many Abelian groups are there of order $2^5 \times 3^6 \times 5^7$?

How many Abelian groups are there of order $2^5 \times 3^6 \times 5^{20}$?

If $5x + 3y = 3$, then find $32^x \times 8^y$.

Which of the following statements are true? Option A: A subring of an integral domain is an integral domain; Option B: A subring of a UFD is a UFD; Option C: A subring of a PID is a PID; Option D: A subring of a Euclidean domain is a Euclidean domain.

Prove that the set of all integers with regular addition and multiplication is an integral domain by showing that it satisfies all the conditions except for the existence of a multiplicative inverse.

Suppose we have a non-empty subset of a group $G$. Prove that this subset is a subgroup if and only if for all $x, y \in H$, $xy^{-1} \in H$.

Show that the centralizer of $H$, defined as $C_H = \{ g \in G \mid gh = hg, \forall h \in H \}$, is a subgroup of $G$.

Show that the conjugate subgroup $G^{-1}HG$, defined as consisting of elements of the form $G^{-1}hG$ for $h \in H$, is a subgroup.

Prove that the function $f(x) = 3x - 2$ is injective.

Prove that the function $f(x) = x^2$ is not injective.

Prove that the function $f(x) = 5x + 2$ is surjective from the reals to the reals.