abstract-algebra

  1. Why are $i$ and $-i$ "more indistinguishable" than $\sqrt{2}$ and $-\sqrt{2}$?

  2. Maximal ideals in a polynomial ring

  3. Understanding the ideal generated by a polynomial
  4. How's GF($2^3$) is calculated in the following picture
  5. On the Lambert series for the Mobius function

  6. Which of these principle ideals is maximal in $\mathbb{Z}[x]$?

  7. $\mathbb Z$ is integrally closed in $\mathbb Q$?
  8. What does Lang mean by "acts in a natural way on $B/\mathfrak{P}$"?

  9. Suppose $I\oplus K$ is a free module, then "$KI\subseteq K\cap I$"
  10. $A$ is a ring and $I$ is an ideal, an element $x \in A$ maps to an invertible element of $A/I$.
  11. If $|H|=4$ and $gH$ has order 3 in $G/H$. Show that $K=H\cup gH\cup g^2H$ forms a subgroup of $G$.

  12. Scalar multiplication well-defined using tensor product universal property

  13. Reference for group cohomology
  14. Generated ideal of an Element with $x=x^{2}$

  15. Questions about cyclic groups (Paar-Pelzl).

  16. Change of basis between $\mathbb{Q}(\sqrt{2}+ \sqrt{3})$ and $\mathbb{Q}(\sqrt{2}, \sqrt{3}) $
  17. Which formal power series can be expressed as a rational fraction?
  18. Show that $\phi: \mathbb{R^{n}}\to \mathbb{R^{m}}$ defined by $\phi(x)=Ax$ is a homomorphism.
  19. composition of rational maps between two curves in Silverman

  20. Why the forgetful functor from $\mathbf{Ab}$ to $\mathbf{Grp}$ does not admit a right adjoint?

  21. What property of modules ensures that module lattice is uniquely complemented?

  22. Question about basis of free modules.

  23. Equivalent definition of free module
  24. If p is a divisor of all coefficients of a polynomial f(x) but $p^2$ does not divide the constant. Show f(x) irreducible over $\mathbb{Q}$

  25. Question on definition of Weyl Group in Brian Hall

  26. Is the following statement correct or incorrect?

  27. Prove or disprove. cycles( 2 4 5 7) and( 3 4 5 6) in s7 are conjugate

  28. Do semirings always require neutral elements?
  29. An unjustified bound in calculation of polynomial resultants
  30. Category theory without codomains?
  31. Show that $4x^2 + 6x + 3$ is a unit in $Z_8[x]$.

  32. Roots of a polynomial of degree $n$ over a ring $R$

  33. Prove isomorphism in infinite-dim condition

  34. Ratio distance similarity transformations: $|\varphi (a)- \varphi (b)| = k|a-b|$

  35. Show that a complex polynomial has a root inside a disk

  36. similarity transformation and algebra
  37. Quotient rings (modules) of a bigger ideal is contained in the quotient ring (modules) of a smaller ideal?

  38. What if $G/M=G/N$.....?

  39. Field generated and fraction field
  40. An exercise about $X$-invariant in automorphism groups.

  41. How to extract square roots of polynomials in several variables?

  42. Prove that the number of distinct zeros of f(x) in a splitting field divides deg f(x)

  43. Show that the following $\phi: \mathbb{R}\to GL_{2}(\mathbb{R})$ is not a homomorphism.
  44. In a module X over R, if $a\in R$ such that $a.x=0$ for all $x\in X$, then $a=0$?
  45. Determine all group homomorphisms of the form $\phi: \mathbb{Z}_{12}\to \mathbb{Z}_{30}$.
  46. What are some mathematical topics that involve adding and multiplying pictures?
  47. $M =\mathbb{C}[x]/(x^2)$ is not a free module over $\mathbb{C}[x]$
  48. Profinite completion of integers

  49. Finitely generated extension of transcendence degree 1

  50. How do I verify if $\phi : \mathbb{Z}_{6} \rightarrow \mathbb{Z}_{36}$ is well-defined?

  51. short Linear Algebra / Orthogonality quiz
  52. How to show $\frac{M_1 \oplus M´}{M_1} \simeq \frac{N_1 \oplus N´}{N_1} $ if we know $M´ \simeq N´$?
  53. Figuring out what $G/H$ looks like if $G=D_3$ and $H=\{R_0,R_{120},R_{240}\}$

  54. Matrix Ring over Semisimple Ring

  55. Show that if $p \equiv 3$ mod $4$ is a prime in $\mathbb{Z}$, then $p$ is a prime in $\mathbb{Z}[i]$
  56. Since 2-cycles (transpositions) are not disjoint, they don't necessarily commute?

  57. Determine Gal (K/P)
  58. Abstract Algebra - define and find kernal of homomorphism from $\mathbb Q[x]$ to $\mathbb Q[\sqrt2]$

  59. Counting elements of a particular order in Dihedral groups

  60. Product of solutions of quadratic equations equal constant term

  61. Generalized Heisenberg algebras
  62. Survived finite subgroup of a Lie group
  63. Group Theory - Abelian group
  64. Cosets: Define a 1-1 and onto function from $Hg \to H$, where $H \le G$

  65. Let $A$ an $R$-ideal, $M$ a submodule generated by $n$ elements, $x \in R$ with $xM \subset AM$. Show that $(x^n+y)M = 0$ for some $y \in A$

  66. Generators of $SL_2({\bf F}_3)$

  67. Prime numbers $p$ and $q$ and possession of normal subgroup of order $p$
  68. Two tori $\mathbb C/L$ and $\mathbb C/L'$ are isomorph if $L=L'$
  69. List all the possible orders for G and calculate $\gamma_G(m)$

  70. If $S<G:=\text{Gal}(E/R)$ is a Sylow 2-subgroup, then $[\text{Fix}_E(S):R] = [G:S]$
  71. Determine the Galois groups of each of these polynomials in $\mathbb{Q}$
  72. Abelian group of type $(n_1, n_2, \ldots, n_t).$

  73. Smallest non-commutative ring with unity

  74. Let $G$ be a group, if $H=\{b\in G\ |\ bab^{-1} \in \langle a \rangle\} $ is H a subgroup of G?
  75. Cyclic group $G$ with generator $ab$.

  76. $\mathbb{R}[Q_8]$ is not a division algebra

  77. Prove isomorphism 0f quotient.

  78. Definition of $K$-conjugacy classes
  79. Constructing a quotient ring in GAP using structure constants
  80. Proof explanation on a group of order $595$ having a normal Sylow $17$-subgroup.
  81. permutations of a 3 object

  82. $R$ is integral domain

  83. Minimal subgroups

  84. Prove that $\alpha^2-3\alpha+1$ is not in the ideal $(5,\alpha+2)$ in $\mathbb{Z[\alpha]}$ where $\alpha=2^{1/3}$

  85. Subgroups of a group of order 44
  86. Localization of Modules over Dedekind domains

  87. What is the ideal of the given ring?
  88. Solve my problem please
  89. Number of polynomials in a finite commutative ring with n elements of degree d and of degree 2 in $R[x_1,x_2]$?

  90. Concise proof for showing $S = (\mathbb{R}\times \mathbb{R})/\{(0, 0)\}$ is closed

  91. What is the motivation of the identity $x\circ(a\circ b)=(((x\circ b)\circ a )\circ b$?

  92. how to check whether it is a Galois extension.

  93. Zeros of power series with polynomially bounded integer coefficients

  94. R is integral domain prove that

  95. The attributes of the defined direct product.

  96. Prove that $\mathbb{Q}^{*} \cong (\mathbb{Z}[x],+)$
  97. If a matrix has a unique left inverse then does it necessarily have a unique right inverse (which is the same inverse)?

  98. Reference request for studying Ring of continious functions

  99. Proof of a member of a set, algebra

  100. Is this the (real) algebraic numbers?