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 Jan 24 comment Why does vector sum $(x_1,x_2)+'(y_1,y_2)=(x_1+2y_1, 3x_2-y_2)$ and $(cx_1,cx_2)$ fail to hold the axiom of vector space? You gave us a formula for it! So it "works" by using it: substitute in actual numbers for the coordinates on the left and the formula on the right tells you what it is. Do you not see how the function $f(a,b) = a+3b$ "works"? Jan 24 comment Why does vector sum $(x_1,x_2)+'(y_1,y_2)=(x_1+2y_1, 3x_2-y_2)$ and $(cx_1,cx_2)$ fail to hold the axiom of vector space? You should not mix ordinary vector addition with the vector operation +'. Check if $(\mathbf x +' \mathbf y) +' \mathbf z = \mathbf x +' (\mathbf y +' \mathbf z)$. That is not what you're doing. Jan 24 comment Is “Generalized functions” by Gelfand published in 5 or 6 volumes? @maojun1998, you now have the title, so how you choose to acquire a copy (for free or by purchasing it) is your business. The download links on the page I mentioned are pretty obvious, but if you don't read Russian then they're irrelevant to you. Jan 24 awarded Nice Answer Jan 23 revised Is “Generalized functions” by Gelfand published in 5 or 6 volumes? deleted 211 characters in body Jan 23 answered Is “Generalized functions” by Gelfand published in 5 or 6 volumes? Jan 23 answered Why is a linear transformation called linear? Jan 5 awarded Nice Answer Dec 30 comment Degree of the splitting field of $X^4-3X^2+5$ over $\mathbb{Q}$ I agree with Tom that the squaring of a sum of 4 terms starts to lead into some nasty algebra, and since we shouldn't bring in ideas that are beyond the scope of a basic Galois theory course or are too computationally intensive for an exam (trace mappings, quartic discriminants, ramification in number fields, etc.) I suspect this is not the kind of solution that would be used on a prelim exam. Dec 30 comment Degree of the splitting field of $X^4-3X^2+5$ over $\mathbb{Q}$ Concerning $\sqrt{11}$ not being in $\mathbf Q(\sqrt{5})$ also, my point was that what I said follows from the lack of a real embedding of $\sqrt{-11}$, not that it is equivalent to this. So your comment is not inconsistent with mine. Dec 30 comment Degree of the splitting field of $X^4-3X^2+5$ over $\mathbb{Q}$ OK, that's a type of "real number trick" that you invariably wind up using with these exam-type problems where the full classification of Galois groups of quartics is unreasonable to bring to bear. Dec 30 comment Degree of the splitting field of $X^4-3X^2+5$ over $\mathbb{Q}$ For the record this requires showing first that $\mathbf Q(\sqrt{5},\sqrt{-11})$ has degree 4 over $\mathbf Q$, which is equivalent to showing $\sqrt{-11} \not\in \mathbf Q(\sqrt{5})$, and that follows from $\mathbf Q(\sqrt{5})$ having a real embedding while $\sqrt{-11}$ of course can't be embedded into $\mathbf R$. Dec 29 comment Degree of the splitting field of $X^4-3X^2+5$ over $\mathbb{Q}$ Since this is meant to be a "basic" question I am wondering if there might be a typographical error, e.g., perhaps the polynomial should be $x^4 - 3x^2 - 5$. I am suggesting this because in these types of exam questions about Galois groups of quartics in $\mathbf Q[x]$ you can usually do some trick about real vs. nonreal roots to make progress in place of using more machinery. With constant term $-5$ such a trick is available (the quartic has real and nonreal roots), but with constant term $5$ this trick doesn't work. Dec 29 comment Degree of the splitting field of $X^4-3X^2+5$ over $\mathbb{Q}$ Where is this question coming from (old qualifying exam?), and what is the assumed background for solving this problem, e.g., are you not allowed to use algebraic number theory? Dec 29 comment Degree of the splitting field of $X^4-3X^2+5$ over $\mathbb{Q}$ It is false that $x_1^2 = x_2^2$: otherwise $x_2 = \pm x_1$, but $x_2$ is neither $x_1$ nor $-x_1$. Dec 29 comment Degree of the splitting field of $X^4-3X^2+5$ over $\mathbb{Q}$ Let $r$ be a root of $f(X)$. Show $[K:\mathbf Q(r)] = 2$ by finding a root $s$ of $f(X)$ that is not in $\mathbf Q(r)$ and checking that all the roots of $f(X)$ are in $\mathbf Q(r,s)$ and that $s$ is quadratic over $\mathbf Q(r)$. Dec 29 revised Why does topology rarely come up outside of topology? added 1763 characters in body Dec 29 answered Why does topology rarely come up outside of topology? Dec 25 comment What is a periodic module Have you heard of the term periodic group? It is synonymous with torsion group: each element has finite order. Nov 10 comment What does it mean to differentiate in calculus? I agree this is a nice example for using the derivative. So many elementary examples boil down to maximizing or minimizing a quadratic polynomial $ax^2 + bx + c$, and for that you can use ideas related to the quadratic formula to see the max/min is at $x = -b/2a$ with no need for derivatives.