Recent content by Adorno

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    Finding the inverse of a polynomial in a field

    Homework Statement (A somewhat similar question to my last one). Let J be the ideal of the polynomial ring \mathbb{Q}[x] generated by x^2 + x + 3. Find the multiplicative inverse of (3x^3 + 3x^2 + 2x -1) + J in \mathbb{Q}[x]/JHomework Equations The Attempt at a Solution I think I need to apply...
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    Factorizing a polynomial over a ring

    So, it's essentially just trial and error?
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    Factorizing a polynomial over a ring

    Homework Statement Factorize x^2 + x + 8 in \mathbb{Z}_{10}[x] in two different waysHomework Equations The Attempt at a Solution I can see that x = 8 = -2 and x = 1 = -9 are roots of the polynomial, so one factorization is (x + 2)(x + 9). Is there a systematic way to find all the factorizations?
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    Sylow Theorems and Simple Groups: Proving Non-Simplicity for Groups of Order 96

    So we define the map phi : G -> S_3 by phi(g) = (left multiplication by g). I can see why this is a homomorphism. (Since left multiplication by g_1g_2 is the same as left multiplication by g_2, then left multiplication by g_1). Is this right?
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    Sylow Theorems and Simple Groups: Proving Non-Simplicity for Groups of Order 96

    So does that action look like (g, n) |-> gn where n is in N_G(H)? Ok, so how about the homomorphism? Do you map an element of g to the action of left multiplication by g on N_G(H)?
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    Sylow Theorems and Simple Groups: Proving Non-Simplicity for Groups of Order 96

    I'm not too good at group actions. What would the action be? And where would the homomorphism come from? If the kernel is normal, then it must be trivial, otherwise there would be a non-trivial normal subgroup of G.
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    Sylow Theorems and Simple Groups: Proving Non-Simplicity for Groups of Order 96

    Homework Statement Prove that no group of order 96 is simple. Homework Equations The sylow theorems The Attempt at a Solution 96 = 2^5*3. Using the third Sylow theorem, I know that n_2 = 1 or 3 and n_3 = 1 or 16. I need to show that either n_2 = 1 or n_3 = 1, but I am unsure how to do...
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    Equivalent conditions on a metric space

    Cool, thanks for your help! I think I've worked out the other parts now.
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    Equivalent conditions on a metric space

    Ok, well my definition of an open set is a set in which every point is an interior point. I think I see now. If there is a nonempty open set, it contains an interior point, and thus there is an open ball within the set, by definition of interior point. So if there are no open balls there can...
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    Equivalent conditions on a metric space

    Ok, I'll see how it goes. My/your/our reasoning proves i -> iii doesn't it? The only problem is the question talks about "open sets", but we were talking about (open) balls. Or does the fact that there are no balls mean that there are no open sets? I'm a bit confused about that
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    Equivalent conditions on a metric space

    So it contains points of A, and therefore can't be disjoint from A. I see. But shouldn't I be proving that ii -> iii and iii -> i?
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    Equivalent conditions on a metric space

    Right. So if you take an open ball disjoint from A, all the points in the ball are limit points... right?
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    Equivalent conditions on a metric space

    Ok, so a limit point of A is a point x in X such that every open ball centred at x contains another point of A, right? I'm not sure where this is going. :-p
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    Equivalent conditions on a metric space

    Yes, that's the definition of dense I'm using. The definition of closure of A, as far as I know, is the union of A with its limit points.
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    Equivalent conditions on a metric space

    Homework Statement Let X be a metric space and A a subset of X. Prove that the following are equivalent: i. A is dense in X ii. The only closed set containing A is X iii. The only open set disjoint from A is the empty set Homework Equations N/A The Attempt at a Solution I can...