Jyrki Lahtonen
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 14h comment Determine all of the monic irreducible polynomials in $\mathbb Z_3 [x]$ of degree $4.$ So it would help i you have found one such irreducible quartic. Can you show us one? Hint: The field $\Bbb{F}_{81}$ is the smalles extension of $\Bbb{F}_3$ that contains a fifth root of unity. Can you get started with that? 14h comment Determine all of the monic irreducible polynomials in $\mathbb Z_3 [x]$ of degree $4.$ (cont'd) I give examples of these techniques in this earlier answer. Unfortunately the list you are expected to build is a bit longer (18 polynomials instead of 6). One extra trick (useless in characteristic two) is that if $p(x)$ is a monic irreducible quartic, then so is $p(-x)$. I don't know if you get all the others by intelligently applying these transformations to a single irreducible. May be, but I don't want check the entire group now :-) 14h comment Determine all of the monic irreducible polynomials in $\mathbb Z_3 [x]$ of degree $4.$ As Eric Wofsey says, some listing will be necessary. You can use the facts (be prepared to prove them, if your teacher insists) that if $p(x)$ is a monic irreducible polynomial, then so are $p(x+1)$ and $p(x-1)$. A slightly more subtle technique is the observation that if $p(x)$ is an irreducible quartic, so is the reciprocal polynomial $x^4 p(1/x)$. These tricks will give you many irreducible polynomials with minimal extra calculations. 15h comment Let $v$ be any vector in a vector space $V$. Prove that: $-v=(-1)v$ Undoubtedly by formal you mean: a deduction based on axioms of a vector space. If so, please give a bit of background. Have you already shown that $0v=\overline{0}$. IOW scalar multiplication of any vector by $0$ will give the zero vector as a result? 19h comment Rate of Koetter-Kschischang Codes $|C|$ is the total number of codewords. If you want to transmit $k$ $q$-ary symbols worth of information, you need to have $q^k$ choices for a codeword. If $|C|=q^k$, then $k=\log_q|C|$. Other than that I'm not at all up to speed with what goes on in network coding. A key differences here is that codewords are subspaces, and the ambient playground is the Grassmannian. This is in sharp contrast to usual error-correcting codes, where the code itself is a subspace, words are individual vectors, and the ambient playground is a bigger vector space together with its Hamming metric. 1d comment Quadratic Equation Manipulation Comments are not for extended discussion; this conversation has been moved to chat. 1d comment Polynomials with even constant term form an ideal in $\mathbb{Z}[x]$ Your work is fine. A nitpicking detail is that that a third, or should I say zeroth, condition for a subset of a ring to be an ideal is that it should be non-empty. In other words you need to also prove that there exists at least one polynomial with an even constant term :-) 1d comment Polynomials with even constant term form an ideal in $\mathbb{Z}[x]$ Reopening now that the question is limited to the question whether such polynomials form an ideal. I agree with user26857 that $I(x)$ is a bit unusual way of denoting this set. Normally that would be reserved for the set of polynomials with ALL their coefficients belonging to that ideal. Whatever :-) Anyway, be cautious about edits that make a major change to the question. Here's the link in case you still need it. 1d comment How to solve system: x_1+x_2+…+x_n=a This works (+1), but only if the characteristic of the underlying field is $>n$. For example in characteristic two the equation $\sum_i x_i^2=a^2$ is a consequence of $\sum_i x_i=a$. As the OP did not specify the field, it is reasonably safe to assume characteristic zero :-) 1d comment Solve $x^2 = 2$ over $F_5$. The solutions of that equation exist only in $\Bbb{F}_{25}$. And outside the subfield $\Bbb{F}_5$. 1d comment Humorous mathematical essays There was once an article about self-referential structures in, IIRC, American Mathematical Monthly. Its list of references had a single item, the article itself. The only theorem of the article boldly stated that Theorem 1. in reference [1] is false. Where they usually place a photo of the author(s), there was a selfie taken with the aid of a mirror, the subject shown taking a picture et cetera... 1d comment Generatos of non-abelian Galois group of order 8. Also, the action of the group on the four zeros is faithful. So the group is isomorphic to a subgroup of $S_4$. Because the order is 8, it is a Sylow 2-subgroup, and hence isomorphic to $D_4$. Nothing wrong with your approach either :-) 2d comment Calculate degree of a number over Q A duplicate of a recent question. Somehow it feels likely that the questions are coming from students taking the same course (rolls eyes). 2d comment Exponential order in Laplace Transform:constructing a function such that it is of exponential order but its derivative is not. There was an error in the definition of $\Delta_{H,A}$. Should be fixed now. The function is supposed to be zero, when $|x|=A/H$. 2d revised Exponential order in Laplace Transform:constructing a function such that it is of exponential order but its derivative is not. added 5 characters in body 2d comment Exponential order in Laplace Transform:constructing a function such that it is of exponential order but its derivative is not. @kilimanjaro: see mrf's (+1) answer for that part. 2d comment Combinatorics problem (Pigeonhole principle). @YvesHalimi: Hard work pays off :-) 2d comment Show that is a maximal ideal of Q[x] Can you find a homomorphism of rings from $\Bbb{Q}[x]$ to $\Bbb{Q}$ such that the kernel is this ideal? 2d answered Exponential order in Laplace Transform:constructing a function such that it is of exponential order but its derivative is not. 2d comment Teaching +, -, *, / to 3 year old kid Here's my story in a relevant thread. In other words I will act on lhf's suggestion. I apologize in advance, if the users of ME.SE feel that this question is not appropriate there/here.