### Questions (21)

 5 Prove that [0,1] is connected. 4 $K/k$ Galois with group $\mathfrak{g}$, $V_{K} \cong V_{k} \otimes K$. Then, $V_{k}$ consists of elements of $V_{K}$ invariant under $\mathfrak{g}$. 2 $F_{\sigma}$ is in the group of automorphisms of $V$. What does it mean to say that $F_{\sigma}$ is defined over $k$? 2 If R is a semisimple ring, then every simple R-module is isomorphic to a simple constituent of R 2 Suppose $B_j = \sum_{i=1}^{r} a_{ij} A_i, j= 1,2,…,t$. How does showing that $B_i's$ are dependent prove that $r \geq n$?

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 3 Can't understand proof for why $f(x) =x^3$, $f: \mathbb{R} \rightarrow \mathbb{R}$ is injective 2 Find the maximum value of $2x + 2\sqrt{x(1-x)}$ if $0 \leq x \leq 1.$ 2 Prove that a square matrix over an algebraic closed field is nilpotent if and only if all their eigenvalues are zero. 2 Show that the set of rank two matrices in $M_{2\times3}(\Bbb R)$ is open. 1 Prove by induction on $n$ that $(1)(2)+(2)(3)+…+n(n+1)={1\over 3}n(n+1)(n+2)$

### Tags (37)

 3 real-analysis × 2 1 proof-explanation × 4 3 functions 1 proof-verification × 3 2 linear-algebra × 2 1 arithmetic 2 algebra-precalculus 1 prime-numbers 2 abstract-algebra 1 calculus

### Bookmarks (33)

 62 How do I tell if matrices are similar? 42 For every infinite $S$, $|S|=|S\times S|$ implies the Axiom of choice 36 Slick proof the determinant is an irreducible polynomial 32 Why maximal atlas 24 Proof that determinant rank equals row/column rank

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 Mathematics 634 rep 33 silver badges1515 bronze badges English Language Learners 121 rep 33 bronze badges Meta Stack Exchange 101 rep MathOverflow 101 rep