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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

2.4 $

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

2.4 $

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

1.92 $

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

3.6 $

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

2.4 $

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

3.6 $

Problem: Are rings \( A \) and \( R \), where \( A \) is a set of complex matrices of the form \( \left(\begin{array}{cc}z & w \\ -\bar{W} & \bar{z}\end{array}\right) \), and \( R \) is a set of real matrices of the form \[ \left(\begin{array}{cccc} x & -y & -z & -t \\ y & x & -t & z \\ z & t & x & -y \\ t & -z & y & x \end{array}\right) ? \]

1.6.18 Fields, Groups, Rings

4.8 $

Problem: Find the rectangle with the largest area inscribed in a circle of radius \( R \).

2.11.5 Function extrema

2.88 $

Problem: Find local extrema of the function \[ z=x^{4}+y^{4}-4 x y \text {. } \]

2.11.6 Function extrema

1.2 $

Problem: Among rectangular parallelepipeds with a given volume \( V \) find the one, having the smallest total surface.

2.11.7 Function extrema

3.6 $

Problem: Let \( e_{1}, \ldots, e_{n} \) be such elements of the center of the ring \( A \) with such 1 , that \( 1=e_{1}+\cdots+e_{n}, e_{i}^{2}=e_{i} \), \( e_{i} e_{j}=0, i \neq j \). Prove that \( A e_{i} \) are two-sided ideals of the ring \( A \) and \( A=A e_{1} \oplus \ldots \oplus A e_{n} \).

1.6.17 Fields, Groups, Rings

6 $

Problem: Solve the problem with moving boundaries: \[ \begin{array}{l} J(x(\cdot))=\int_{0}^{T_{0}}\left(\dot{x}^{2}-x\right) d t \rightarrow \text { extr }, \quad x(0)=0, \\ x\left(T_{0}\right)=\xi . \end{array} \]

4.21 Variational calculus

4.8 $

Problem: Solve the problem of Bolza: \[ \int_{1}^{\mathrm{e}} 2 \dot{x}(t \dot{x}+x) d t+3 x^{2}(1)-x^{2}(e)-4 x(e) \rightarrow \text { extr. } \]

4.22 Variational calculus

6 $

Problem: Solve the simplest problem of the classical calculus of variations: \[ \begin{array}{l} \int_{0}^{T_{0}}\left(2 \dot{x}^{2}+x^{2}-4 x \sin t\right) d t \rightarrow \text { extr }, \\ x(0)=0, \quad x\left(T_{0}\right)=\xi . \end{array} \]

4.23 Variational calculus

4.8 $

Problem: Solve the problem with moving boundaries: \[ \begin{array}{l} \int_{0}^{1}\left(\frac{\dot{x}_{1}^{2}+\dot{x}_{2}^{2}}{2}-x_{1} x_{2}\right) d t \rightarrow \text { extr }, \\ x_{1}(0)=x_{2}(0)=1 \end{array} \]

4.24 Variational calculus

3.6 $

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