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2h
comment How to find minimal polynomial of primitive element (field theory)
+1 The somewhat surprising part is that the answer is completely independent of the minimal polynomial of the given primitive element.
3h
comment A problem in finite field and its order
You should search the site, for this question has been asked and answered many times. Restrict the search to the finite-field tag, and you should get lucky sooner rather than later!
3h
comment Generator polynomial creates a 127 bit sequence
Tyrone: I think that the polynomial $G(D)$ is used as a recurrence relation: $$s_{n+7}+s_{n+4}+s_n=0$$ for all $n$. From this we can solve $$s_{n+7}=s_{n+4}+s_n.$$ So if the initial segment is $s_0=s_1=s_2=\cdots=s_6=1$, then $s_7=s_4+s_0=0$, $s_8=s_5+s_1=0$, $s_9=s_6+s_2=0$, $s_{10}=s_7+s_3=1$ et cetera, which seems to fit (assuming that the earliest bits of the sequence are at the right end). I always recheck little-endian vs. big-endian details here, because I tend to forget them. Also, sometimes people use reciprocal polynomials. I won't get into details, but those work equally well.
6h
comment Unit group of $\mathbb{Z}C_m$
It seems to that the analogues of the cyclotomic units $(1-c^k)/(1-c)$, $\gcd(k,m)=1$, at least will be there. I don't know whether they generate the group of units? IIRC they generated a finite index subgroup of the units in the case $\Bbb{Z}[\zeta_m]$, but I'm too ignorant :-(.
1d
revised What is the decomposition of $x^4+x^3+x^2+x+1$.
added 3 characters in body; edited title
1d
revised What is the decomposition of $x^4+x^3+x^2+x+1$.
edited title
1d
revised What is the decomposition of $x^4+x^3+x^2+x+1$.
deleted 3 characters in body; edited title
1d
comment What is the decomposition of $x^4+x^3+x^2+x+1$.
You should specify the question a bit more. If you are looking for factorizations with integer or rational coefficients, then you can use Eisenstein. Did's hint is based on the assumption that you want factors with real coefficients (or what amounts to the same, factors with coefficients in $\Bbb{Q}(\sqrt5)$). Some posters apparently think that you want a factorization with complex coefficients. Don't leave us guessing :-)
1d
comment Factorisation of large polynomials and Galois theory
The emphasis on that consequence of Galois theory is on a general polynomial. We can surely write the zeros of, say $x^5-2$, using radicals. Also, we have no problems whatsoever jotting down the lowest degree polynomial with integer coefficients that has a number given in terms of radicals, say $\root5\of2+\sqrt2$, as one of its zeros. But it is always nice to have inventive methods for constructing examples of many kinds. Can you tell us more? Anyway, a specific family of polynomials with zeros expressible in terms of radicals does not contradict Galois theory. No matter its degree.
2d
revised Exact value of sin (θ/2) if cos θ = 3/5
edited title
2d
comment Exact value of sin (θ/2) if cos θ = 3/5
@TheChaz2.0: The question was edited. Originally it stated $\cos\theta=4/5$
2d
comment Generator polynomial creates a 127 bit sequence
Tyrone, I'm sure that's not the initial state. If you put an LFSR to all zeros state initially, it stays there. Nevertheless, the feedback polynomial is primitive, so you do get an $m$-sequence: a sequence of bits repeating with a period of length 127 such that all 7 bit combinations occur as conseuctive bits (with the exception 7 zeros).
2d
comment Roots of $X^2-z$ in $2$-adics
The list of "Related" questions in the right margin has a lot of helpful material. This question in particular is useful to you.
2d
comment Group ring of a cyclic group over a finite field
See this answer for an explanation of what happens when $\gcd(n,p)=1$.
2d
comment Group ring of a cyclic group over a finite field
This is not true. The splitting into summands depends on whether $p$ is a factor of $n$ and (more importantly) what is the smallest extension field of $\Bbb{F}_p$ that contains $n$th roots of unity (when $p$ and $n$ are coprime). Anyway, the block of the trivial representation will always be there, so the group ring cannot be a field.
2d
comment $f(x,y)=x^3+3xy^2-2y^3$. Find all unit vectors, if any, such that $f_u(0,1)=\frac{6}{5}$
$u$ is a unit vector so you get another equation $u_1^2+u_2^2=1$
May
21
answered Roots of $X^{l-1}+1$ in a quadratic extension $F_q$, $q=l^2$
May
20
awarded  Nice Answer
May
20
comment How to find inverse of generator of a finite field?
What Robert Israel writes (+1). If you want to learn a bit more about doing arithmetic operations in this field take a look at this Q&A I prepared for referrals like this. Your $g$ is denoted $\gamma$ there, so $g^{-1}=g^{14}$ is $\gamma^{14}=\gamma^3+1=1001$.
May
20
comment How to compute $\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$?
Are you familiar with methods from complex analysis? Residues and such?