Finite fields Definition and 35 Threads

Hello all, I have here an excerpt from Wikipedia about the discrete Fourier transform: My question(s) are about the red underlined part. 1.) If ##n## divides ##p-1##, why does this imply that ##n## is invertible? 2.) Why does Wikipedia take the effort to write out the ##n## as ##n =...

I am going to give up a bit more on the given problem. We start with polynomial ## x^27 -x ## over GF(3)[x] and we factorize it using a well known theorem it turns out it factorises into the product of monic polynomials of degree 1 and 3, 11 of them all together. We then choose one of those...

Summary: countability, topological vector spaces, continuity of linear maps, polynomials, finite fields, function theory, calculus1. Let ##(X,\rho)## be a metric space, and suppose that there exists a sequence ##(f_i)_i## of real-valued continuous functions on ##X## with the property that a...

I thought i understood the theorem below: i) If A is a matrix in ##M_n(k)## and the minimal polynomial of A is irreducible, then ##K = \{p(A): p (x) \in k [x]\}## is a finite field Then this example came up: The polynomial ##q(x) = x^2 + 1## is irreducible over the real numbers and the matrix...

I am reading David S. Dummit and Richard M. Foote : Abstract Algebra ... I am trying to understand the example on Finite Fields in Section 13.5 Separable and Inseparable Extensions ...The example reads as follows: My questions are as follows: Question 1In the above text from D&F we read the...

I am reading David S. Dummit and Richard M. Foote : Abstract Algebra ... I am trying to understand the example on Finite Fields in Section 13.5 Separable and Inseparable Extensions ...The example reads as follows: My questions are as follows: Question 1In the above text from D&F we read the...

I am unsure of my approach to Exercise 2 Dummit and Foote, Section 13.2 : Algebraic Extensions .. I am therefore posting my solution to the part of the exercise dealing with the polynomial g(x) = x^2 + x + 1 and the field F = \mathbb{F}_2 ... ... Can someone please confirm my solution is...

I am reading Beachy and Blair's book: Abstract Algebra (3rd Edition) and am currently studying Section 6.5: Finite Fields, I need help with a statement of Beachy & Blair in Example 6.5.2 on page 298. Example 6.5.2 reads as follows:https://www.physicsforums.com/attachments/2858In the above...

I am reading Beachy and Blair's book: Abstract Algebra (3rd Edition) and am currently studying Theorem 6.5.7. I need help with the proof of the Theorem. Theorem 6.5.7 and its proof read as follows:In the above proof, Beachy and Blair write: By Lemma 6.5.4, the set of all roots of f(x) is a...

I am reading Beachy and Blair's book: Abstract Algebra (3rd Edition) and am currently studying Proposition 6.5.5. I need help with the proof of the proposition. Proposition 6.5.5 and its proof read as follows: In the proof of Proposition 6.5.5 Beachy and Blair write: " ... ... Since F is the...

I am reading Beachy and Blair's book: Abstract Algebra (3rd Edition) and am currently studying Theorem 6.5.2. I need help with the proof of the Theorem. Theorem 6.5.2 and its proof read as follows:In the conclusion of the proof, Beachy and Blair write the following: " ... ... Hence, since F...

i am studying finite fields and trying to get an idea of the nature of finite fields. In order to achieve this understanding I am bring to determine the elements and the addition and multiplication tables of some finite fields of small order. For a start I am trying to determine the elements...

Homework Statement Let w be a primitive n-th root of unity in some finite field. Let 0 < k < n. My question is how to rationalize [\tex]\dfrac{1}{1 + w^k}[\tex]. That is, can we get rid of the denominator somehow? I know what to do in the case of complex numbers but here I'm at a loss...

Homework Statement Given some ElGamal private key, and an encrypted message, decrypt it. Homework Equations Public key (F_q, g, b) Private key a such that b=g^a Message m encrypted so that r=g^k, t=mb^k Decrypt: tr^-a = m The Attempt at a Solution My problem is finding r^-a...

Homework Statement Let p be an odd prime. Then Char(Z_p) is nonzero. Prove: Not every element of Z_p is the square of some element in Z_p.Homework Equations The Attempt at a Solution I first did this, but i was informed by a peer that it was incorrect because I was treating the congruency as...

Homework Statement Let q=pm and let F be a finite field with qn elements. Let K={x in F: xq=x} (a) Show that K is a subfield of F with at most q elements. (b) Show that if a and b are positive integers, and a divides b, then Xa-1 divides Xb-1 i. Conclude that q-1 divides...

Hi, yet another question regarding polynomials :). Just curious about this. Let f(x), g(x) be irreducible polynomials over the finite field GF(q) with coprime degrees n, m resp. Let \alpha , \beta be roots of f(x), g(x) resp. Then the roots of f(x), g(x), are \alpha^{q^i}, 0\leq i \leq n-1...

Hi, all: Say we have a bare-bones Vector Space v, i.e., V has only the basic vector space layout; no inner-products, etc., over a finite field . I think then , we can still define a line in V as the set {fvo: vo in v, f in F}, i.e., as the set of all F-multiples of a fixed...

Homework Statement If A=<1+i> in Z[i], show that Z[i]/A is a finite field and find its order Homework Equations The Attempt at a Solution Not sure where to start... Z[i]/A = {m+ni + A, m, n integers} ? is that right? And I don't know what else to do.

Hi, I am taking a class in Linear Algebra II as a breadth requirement. I have not studied Algebra in a formal class, unlike 95% of the rest of the class (math majors). My LA2 professor mentioned the following fact in class: "The number of elements of a finite field is always a prime power and...

Homework Statement Solve 3x + 50 = 11 in F53 Homework Equations Extended Euclidean AlgorithmThe Attempt at a Solution To find 3-1 mod 53 using the euclidean algorithm: gcd(53,3) 1)53/3 = 17 + 2R 53 = 17 * 3 + 2 2 = 53 - 17 * 3 3/2 = 1 + 1R 3 = 1 * 2 + 1 1 = 3 - 1 * 2 = 3 - 2Now...

Homework Statement This question is in two parts and is about the field F with q = p^n for some prime p. 1) Prove that the product of all monic polynomials of degree m in F is equal to \prod (x^(q^n)-x^(q^i), where the product is taken from i=0 to i=m-1 2) Prove that the least common multiple...

Homework Statement Show that a finite field of p^n elements has exactly one subfield of p^m elements for each m that divides n. Homework Equations If F \subset E \subset K are field extensions of F , then [K:F] = [E:F][K:F] . Also, a field extension over a finite field of p elements...

Trying to do i) and iii) on this past exam paper For part i) I'm pretty stumped I've said that the possible roots of the polynomial are +- all the factors of T In particular rt(T) needs to be a factor of T but this can't be possible? Doesn't sound too good but its the best I've got. Part...

Homework Statement Assuming the mapping Z --> F defined by n --> n * 1F = 1F + ... + 1F (n times) is a ring homomorphism, show that its kernel is of the form pZ, for some prime number p. Therefore infer that F contains a copy of the finite field Z/pZ. Also prove now that F is a finite...

F3 is a finite field with 3 elements and V is a vector space of n-tuples of elements from F3. Is there a way to calculate the maximum number of elements in a subset S of V, such that for no three elements a,b,c in S a+b+c=0? Or in other words, no three elements in S are on the same line...

Homework Statement Suppose that m = 1 mod b. What integer between 1 and m-1 is equal to b^(-1) mod m? The Attempt at a Solution m = 1 mod b means that: m = kb + 1 for some integer k Let x be the inverse of b mod m, note: x exists since b and m must be coprime due to the previous...

Is it true as it is for finite fields of order p^1, that the number of primitive roots of fields of order p^n is the euler totient of (P^n-1)? If not is there a different rule for the number?

Hi, We recently started analyzing linear machines using matrix algebra. Unfortunately, I haven't had much exposure to operating in finite fields aside from the extreme basics (i.e. the definitions of GF(P)). I can get matrix multiplication/addition, etc. just fine, but it's when finding the...

It's hard to find the proofs of these theorems. Please help me... Thanks! Theorem 1: Let V be a vector space over GF(q). If dim(V)=k, then V has \frac{1}{k!} \prod^{k-1}_{i=0} (q^{k}-q^{i}) different bases. Theorem 2: Let S be a subset of F^{n}_{q}, then we have dim(<S>)+dim(S^{\bot})=n.

Hello, everyone, i am a newbie here. I am currently taking a modern linear algebra course that also focus on vector spaces over the fields of Zp and complex numbers. Since i am not familiar with typing up mathematics using tex or anything so that i can post on the forums, i will use the...

I have the definition that if F is a finite field then a \in F is a primitive root if ord(a) = |F|-1. Now what I don't understand is how exactly are there \phi(|F|-1) primitive roots? (Note: This material is supposed not to use any group theory.)

Can someone explain to me why the following is true (ie, show me the proof, or at least give me a link to one): Over the field Zq the following polynomial: x^q^n-x is the product of all irreducible polynomials whose degree divides n Thanks.

I am trying to figure out the effect of a field automorphism on a field with a non prime subfield. Say for example F_{2^{29}}, F_{2^{58}} and F_{2^{116}} Let \alpha \in F_{2^{58}}\F_{2^{29}} Under {\sigma}^{i}, 1 \le i \le 58 do we get any case where \alpha becomes an element of...

Does anyone know if this is true and if so where they know it from? Given a polynomial over the integers there exists a finite field K of prime order p, such that p does not divide the first or last coefficient, and the polynomial splits over K. I realize this could be considered an...