Fields, Groups, Rings/Fields, Groups, Rings

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Problem: Construct non-isomorphic groups with 3-element base set.

1.6.1 Fields, Groups, Rings

2 $

Problem: Find the matrix centralizer \( A=\left(\begin{array}{ll}3 & -4 \\ 1 & -2\end{array}\right) \) in \( G L(2, R) \), where \( G L(n, F) \) is a group of non-degenerate matrices of size \( n \times n \) over the field \( F \), with the operation of matrix multiplication.

1.6.2 Fields, Groups, Rings

3.01 $

Problem: Prove that the groups \[ \left\langle R \backslash\{0\}, \cdot,{ }^{-1}\right\rangle \text { and }\left\langle R_{>0}, \cdot,{ }^{-1}\right\rangle \] are not isomorphic.

1.6.4 Fields, Groups, Rings

2 $

Problem: Find the order of each element in the group of 6th roots of 1 .

1.6.5 Fields, Groups, Rings

3.76 $

Problem: On the set \( A=\left\{\langle a, b\rangle \mid a, b \in R, a^{2}+b^{2} \neq 0\right\} \) the operation \( \langle a, b\rangle *\langle c, d\rangle=\langle a c-b d, a d+b c\rangle \) is defined. Define the transition operation \( { }^{-1} \) to the inverse element so that the set \( \left\langle A, *,{ }^{-1}\right\rangle \) is a group.

1.6.3 Fields, Groups, Rings

5.01 $

Problem: Is the mapping \( f \) of the group \( \langle\mathbb{Z} / 4 \mathbb{Z},+,-\rangle \) to the group \( \langle\mathbb{Z},+,-\rangle \), homomorphic such that a) \( f(\overline{2})=2 \); b) \( f(\overline{3})=3 \).

1.6.6 Fields, Groups, Rings

3.01 $

Problem: Prove that the factor group \( \mathbb{R}^{*} / \mathbb{Q}^{*} \) is not cyclic.

1.6.7 Fields, Groups, Rings

3.76 $

Problem: Let \( (M, \cdot) \) - be a group, and \( \alpha \) and \( \beta \) - mappings from \( M \) to \( M \). Let's define an operation * on \( M \) by the rule: \( x * y=\alpha(x) \beta(y) \forall x, y \in M \). Prove that the conditions are equivalent: (1) \( (M, *) \) - a group; (2) \( \exists a, b \in M: \alpha(g)=g a, \beta(g)=b g \quad \forall g \in M \).

1.6.9 Fields, Groups, Rings

6.26 $

Problem: Prove that \( \mathbb{Q}(p) / \mathbb{Z}^{+} \cong \mathbb{Z}_{p^{\infty}} \).

1.6.10 Fields, Groups, Rings

4.26 $

Problem: Let \( R \) be a commutative ring with 1 , such that \( R[x] \) is a principal ideal ring. Prove that \( R \) is a field.

1.6.11 Fields, Groups, Rings

2.51 $

Problem: Is \( \left\{\left.\frac{m}{n} \right\rvert\, m, n \in \mathbb{Z} ; n \notin p \mathbb{Z} ; m \in p \mathbb{Z}\right\} \) ideal in the \( \mathbb{Q}_{p} \), ring of all rational numbers, represented in the form of a fraction with a denominator not divided into a simple number \( p \) ?

1.6.12 Fields, Groups, Rings

2.51 $

Problem: Which of the numbers \( 1-3 i, 3+i, 3-i \) are decomposable in the ring \( \mathbb{Z}[i] \) ?

1.6.13 Fields, Groups, Rings

2 $

Problem: Let \( A \) be the integral ring, and \( B \) - subring of the ring \( A \), such that \( e_{A} \in B \). Is it true that if \( A \) is a factorial ring, then \( B \) is a factorial ring too?

1.6.14 Fields, Groups, Rings

3.76 $

Problem: Express the symmetric polynomial \[ \left(x_{1} x_{2}+x_{3} x_{4}\right)\left(x_{1} x_{3}+x_{2} x_{4}\right)\left(x_{1} x_{4}+x_{2} x_{3}\right) \] in the ring \( \mathbb{Q}\left[x_{1}, x_{2}, x_{3}, x_{4}\right] \) in terms of elementary symmetric functions.

1.6.15 Fields, Groups, Rings

2.51 $

Problem: Calculate the reduced Gröbner basis of the ideal \[ I=\left(x^{2}+y^{2}+z^{2}-1, x^{2}+z^{2}-y, x-z\right) \] in \( \mathbb{Q}[x, y, z] \) with lex-ordering \( x>y>z \).

1.6.16 Fields, Groups, Rings

3.76 $

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