Let f(x) be the mean of five numbers: 4, 5, 7, 9 and x. Let g(x) be the median of the same numbers.
For how many values of x, a real number, is f(x) = g(x)?
I only got 2. x = 0, 10
There are 3 though. Perhaps someone can help me find the other one.
How do you find an infinite number of units of \mathbb{Q}(\sqrt{21}) using the \sqrt(21)? I saw one example using continued fractions but do not know how to apply it in this case. I do have the periodic form of the continued fraction of \sqrt(21).
So you see it all over the place, \mathbb{Q}(\sqrt{-5}) is not a UFD by finding an element such that it has two distinct prime factorizations...but what about showing that \mathbb{Q}(\sqrt{5}) is a UFD?
I'm only concerned with this particular example, I might have questions later on regarding a...
Ok, the book I'm reading states Gauss's lemma as such:
If f(x) is a monic polynomial with integral coefficients that factors into two monic polynomials with coefficients that are rational, f(x) = g(x)h(x), then g(x), h(x) \in \mathbb{Z}[x].
Now one of the exercises says to prove that:
If...
Ok I need to know which is the right answer for evaluating the continued fraction \langle 1, 2, 1, 2, \ldots \rangle?
Here's my work:
x = 1 + \frac{1}{2+x} \Rightarrow x^2 + x - 3 = 0 and by quadratic formula, we get x = \frac{-1 \pm \sqrt{13}}{2} but we only want the positive root so I...
Would like to show \sum_{d \mid n} \mu (d) \sigma_0 (d) = (-1)^{\omega (d)}.
This proof is just left out of text I'm looking at and I can't seem to piece how F(n/d) = \sigma_0 (d), where F(x) = \sum_{s \mid x} f(x).
If f is completely multiplicative, then \sum_{d \mid n} f(d) is completely multiplicative is not true. There must be an easy counterexample for this yet I cannot come up with one.
Homework Statement
Show x^2 + (p+1)/4 \equiv 0 (\mod p) where p \equiv 3 (\mod 4) and p is prime is not solvable.
Homework Equations
Legendre's and Jacobi symbol, congruences
The Attempt at a Solution
Noticing that x^2 \equiv -(k+1) (\mod p) when p = 4k + 3 ?
Now (-1/p)(k+1/p) should tell...
So upon introduction to Euler's phi function, we can see that \phi (1) = 1 and \phi (2) = 1, where it turns out that these are in fact the only numbers in N that map to 1. Now what I'm wondering is if there is some general way to find the inverse image of numbers in the image of phi?
Also...
If a and 77 are relatively prime, show that for positive integers n, a^(10^n) modulo 77 is independent of n.
I think I don't understand what this statement is asking. a^(10^n) modulo 77 independent of n means that a^(10^n) modulo 77 is always going to be the same or something?
I was just browsing for some small problems the other day and came across this problem and I am unsure if it should be obvious and have a quick answer. In any case, I couldn't figure it out.
If p is a prime that divides a - b, then show that p^2 divides a^p - b^p, where a, b \in \mathbb{Z}.
So this is supposed be an introductory problem for tensor products that I was trying to do to verify I am understanding tensor products....turns out I'm not so much
Show that M_n(K) is isomorphic as an F-algebra to K \otimes_F M_n(F) where F is a field and K is an extension field of F and...
Let f, g \in \mathbb{Z}[x, y, z] be given as follows: f = x^8 + y^8 + z^6 and g = x^3 +y^3 + z^3. Express if possible f and g as a polynomial in elementary symmetric polynomials in x, y, z.
Professor claims there is an algorithm we were supposed to know for this question on the midterm. I...
Let R be a ring with 1_R. If M is an R-module that is NOT unitary then for some m \in M, Rm = 0.
I'm pretty sure Rm = \{ r \cdot m \mid r \in R \}. While M being not unitary means that 1_R \cdot x \neq x for some x \in M. I'm thinking this problem should be an obvious and direct proof but I...
This stuff is killing me...
Let K \leq M \leq L be fields such that L is galois over M and M is galois over K. We can extend \phi \in G(M/K) to an automorphism of L to show L is galois over K.
I need help filling in the details in why exactly L is galois over K.
This statement was made in my class and I'm trying still to piece together the details of it...
We say that some rational polynomial, f has a Galois group isomorphic to the quaternions. We can then conclude that the polynomial has degree n \geq 8.
I have a few thoughts on this and I might...
There was a part c and d from a question I couldn't answer.
Let R = \{ a/b : a, b \in \mathbb{Z}, b \equiv 1 (\mod 2) \}.
a) was find the units, b) was show that R\setminus U(R) is a maximal ideal. Both I was successful. But
c) is find all primes, which I believe i only found one....the...
Homework Statement
Show that {p \choose k} = \sum^{k+1}_{i=1} {p-i \choose p-k-1} where \forall k < p \in \mathbb{Z} and p a prime.
Homework Equations
This is part (b) to a problem. Part (a) is showing that 1 + x + x^2 + \cdots + x^{p-1} is irreducible in \mathbb{Q}[x].
The Attempt...
Homework Statement
Assuming R is an integral domain. If the polynomial ring of one variable, R[x], is a unique factorization domain, then R is a unique factorization domain.
The Attempt at a Solution
Should be straightforward...so much so that I don't know how to start...probably with...
I saw in an application of Sylow's theorems, it said we have something like a group of order 28 = 2^2 x 7, so we have either 1 or 7 sylow 2-subgroup. Assuming we have 7 sylow 2-subgroups, then we have 21 non-identity elements and the identity, and we have 1 sylow 7-subgroup, blah blah blah...
Homework Statement
G is a finite p-group, show that G/ \Phi (G) is elementary abelian p-group.
Homework Equations
\Phi (G) is the intersection of all maximal subgroups of G.
The Attempt at a Solution
By sylow's theorem's we have 1 Sylow p-subgroup which is normal, call P. Then the order...
Homework Statement
G acts transitively on S and let H be the stabilizer of s. Show that the normalizer of H, call it N, acts transitively on the fixpoints of H, call it F, where s is some element in S.
Homework Equations
Two different ways of showing this:
Either we show the orbit for any...