It's a well known fact that Euclidean domains are principal ideal domain by taking the element in a certain ideal with smallest \varphi value. Yet the converse is false, and one counterexample just lies among our familiar algebraic integer rings, \……

题目背景 为证一个非零吃大师 题面 Assuming AC, prove that for a integral domain R, if any nonzero prime ideal P contains a nonzero prime (p), R is UFD. 先想再看提示哦 提示 Let S be the set of elements that are prime factori……

This proof is due to Choudum. A sequence of nonnegative integers (d_1,...,d_n) is called graphic if there is a simple graph G with V(G)=\lbrace v_1,...,v_n\rbrace and d(v_i)=d_i. It's very easy to show that if a nonincreasing sequence, i.e. d_1\ge ……

飞机上半昏迷状态下以为无法调整, 暴力算出来的搞笑解答. 非常具有统计色彩 (nod) 题面 A simple graph G is called complete k-partite if there's a partion of V(G): A_1,..., A_k s.t. for all i there's no edge with both vertices in A_i , and for any v\in A……

It's a well-known fact that the alternating groups, A_n, are simple for n\neq 4, with small cases like A_1, A_2, A_3 very easy to verify. Yet most proofs of n\geq 5, including Hungerford's, relies heavily on 'magical' permutation composition, which……

This proof is due to Arturo Magidin. In category theory, as we don't consider the 'internal structure' of objects, many properties of maps are defined in an abstract way using interaction with other morphisms. 'Epimorphism' is such an analog of sur……

题目背景 证明连续性时缩放水平哈哈了(sweatgrinning) 题面 f is a nonconstant holomorphic function on a region \Omega_1 and g is defined on f(\Omega_1). Prove that if h=g\circ f is holomorphic, then so is g. 先想再看提示哦 提……

题目背景 完整题解来自ox帅气lecturer, 我只能证明F<\mathbb C的情况 ;( ;( ;() 题面 V是域F上的一个线性空间, \dim V=n, 且有T\in L(V). 现取\lambda_1,...,\lambda_n\in F两两不等, 且任意\lambda_i都不是T的特征值. 试证明存在\alpha_1,...,\alpha_n\in F使得 \……

题目背景 人见人爱的代数小题 :] 惊人的是竟然在Hungerford II.2.8.9才出现, 还标了名字 😮 题面 A^ * is a normal subgroup of A, B^ * is a normal subgroup of B. Prove that: (i) A^ *(A\cap B^ *) is a normal subgroup of A^ *(A\cap B) (ii) B^ *(A^ *\cap B) i……

题目背景 本题竟然与xgn目前正在学习的序数有关, 真是可喜可贺!(crim boi在thu里有包含序数的课程吗?) 题面 If you don't remember what a topology is, please refer to this passage. Preliminary: Open sets in \mathbb R are just union (can be infinite) of som……