# What is Finite fields: Definition and 35 Discussions

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod p when p is a prime number.
Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory.

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1. ### I Question on invertibility in finite fields

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 =...
2. ### I Looking for a creative or quick method for finding roots in GF(p^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...
3. ### Challenge Math Challenge - August 2021

Summary: countability, topological vector spaces, continuity of linear maps, polynomials, finite fields, function theory, calculus 1. 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...
4. ### I Finite fields, irreducible polynomial and minimal polynomial theorem

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...
5. ### MHB Existence and Uniqeness of Finite Fields .... Example from D&F ....

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 1 In the above text from...
6. ### I Existence and Uniqeness of Finite Fields ....

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 1 In the above text from...
7. ### MHB Confirm/Critique Solution to Dummit and Foote, Section 13.2: Exercise 2

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...
8. ### MHB Finite Fields - F_4 - Galois Field of Order 2^2

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/2858 In the...
9. ### MHB Existence of Finite Fields with p^n Elements

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)...
10. ### MHB Characterization of subfields of finite fields - Beachy & Blair - Proposition 6.5.5

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...
11. ### MHB Finite Fields and Splitting Fields

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...
12. ### MHB Nature and character of Finite Fields of small order

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...
13. ### Rationalizing fractions over finite fields

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...
14. ### Polynomial finite fields; ElGamal decryption

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...
15. ### Working with finite fields of form Z_p

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...
16. ### Finite Fields Homework: Showing K has q Elements

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...
17. ### Irreducible Polynomials over Finite Fields

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...
18. ### Generalization of Lines, Planes (Finite Fields)

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...
19. ### Finite Field Order of Z[i]/A

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.
20. ### Exploring Prime Powers in Finite Fields

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...
21. ### Solve for X in Finite Fields

Homework Statement Solve 3x + 50 = 11 in F53 Homework Equations Extended Euclidean Algorithm The 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 - 2...
22. ### Finite fields and products of polynomials

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...
23. ### Finite Fields in Algebra

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...
24. ### Irreducibility and finite fields

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...
25. ### Finite Fields and ring homomorphisms HELP

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...
26. ### Collinear vectors over finite fields

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...
27. ### Units for finite fields

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...
28. ### Number of primative roots in finite fields of order p^n

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?
29. ### Matrix algebra over finite fields

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...
30. ### Help me with these proofs (vector spaces over finite fields)

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.
31. ### Questions concerning Finite Fields (Basic Abstract Algebra)

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...
32. ### Counting Primitive Roots in Finite Fields without Group Theory

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.)
33. ### Irreducible polynomials over finite fields

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.
34. ### Automorphisms of Finite Fields

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...
35. ### Splitting Polynomials Over Finite Fields: Fact or Fiction?

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...