# Tagged Questions

Finite fields are structures arising in abstract algebra. The order of a finite field is always a prime power, and for each prime power $q$ there is a single isomorphism type. It is usually denoted by $\mathbb{F}_q$ or $\operatorname{GF}(q)$.

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### Dependence of algebraic elements in a finite field

Lets work over the finite field $\mathbb{F}_p$ for a prime $p$. Consider a monic irreducible polynomial $f(X)=X^3+aX^2+bX+c$ in $\mathbb{F}_p[X]$. Let $x$ be a root of $f(x)=0$ (say, in the closure of ...
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### Extending a code by adding a parity check

Let $C$ be a $[n,k,d]_2$ code where $d$ is odd. It is known that you can construct a $[n+1,k,d+1]_2$ code by adding a column $\boldsymbol{c}_{n+1}$ to the codebook matrix where each element contains ...
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### How to represent $GF(q)$ element in Matlab [on hold]

Everyone, I am trying to simulate in Matlab the Galois field $GF(q)$ where $q$, for example, is $8$. I know the elements of $GF(8)$ are $0,1,\alpha,\alpha^2,\alpha^3,\ldots,\alpha^6$. Also, we can ...
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### Counting GF($q=8$) matrices with a certain property

Let us denote by $\boldsymbol{v}_i$ the columns of an $m \times n$ GF($8$) matrix. The field elements are enumerated $\{0,1,2,...,q-1\}$. To define the arithmetic operations between field elements, we ...
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### Find roots of polynomial in a finite field

I need to build a field $L$ of 121 elements and find how many roots polynomial $g=x^9-1$ has in $L$. Then to find all these roots. So, $121=11^2$ this is power of prime. We can build finite field of ...
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### What is the meaning of $\Bbb{Q}[x]/f(x)$?

I am very confused with the meaning of $\Bbb{Q}(x)/f(x)$. Does it mean the set of all polynomials modulo $f(x)$? If it does then how can we say that $\Bbb{R}[x]/(x^2+1)$ is isomorphic to set of ...
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### Odd degree of extension field in Couveignes Square root method

I was reading the Couveignes method to find square root for Number Field Sieve(reference here page 4 first line). It says that for this method the degree of extension K/Q must be odd so that Norm(-x)=-...
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### A question on finite fields.

Suppose you have a finite field $\mathbb{F}$, where $|\mathbb{F}|=p^n$ and $p$ is a prime number. Also suppose that $f(x)=x^2+b\in \mathbb{F}[x]$ is an irreducible polynomial over $\mathbb{F}$ and $r$ ...
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### Irreducible polynomial of every degree over finite field

The existence of polynomials in title has been asked as a problem on MathStack many times; some answers were using existence of finite bigger fields, and some answers concern Mobius function with ...
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### Reducibility of $x^q -x -1$ in $\mathbb{F}_{q}$

I came across the following excercise and do not know how to go about this. Given the polynomial $x^q -x -1$ in $\mathbb{F}_{q}$. Consider $q=8$. Show this polynomial is reducible by considering an ...
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### Nilpotence and conjugacy in $M(p,\mathbb F_p)$

I have to solve the following problem: Characterize matrices $X\in M(p,\mathbb F_p)$ (note that $p$ is the dimension and the characteristic of the field) such that there exists $Y$ with the ...
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### Why are these two calculations in $GF(2^5)$ not equal [closed]

I have a quick question about Galois fields, since there seems to be something I have misunderstood way back in university. Addition and subtraction in Galois fields are both done using XOR ...
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### Why are finite field elements polynomials

Finite fields are split up into two parts. Prime fields, arithmetic is simply mod p.A prime fields takes the form $GF(p)$, where $p$ is prime. Why for extension fields, eg, of the form $GF(p^m),m>1$...
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### Show $L$ can be extended to $M$ with $M/F$ cyclic

Suppose that $F$ has characteristic $p$ and $L/F$ is a cyclic extension of degree $p$. I'm trying to show that $L$ can be extended to $M$ where $F\subset L\subset M$ with $M/F$ cyclic of degree $p^2.$ ...
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### Splitting field in relation to finite fields

I'm trying to prove that the subfields of a Galois Field $GF$ of order $p^n$ are isomorphic to a Galois field of order $p^r$ where $r|n$, and that there exists a unique subfield for each such $r$. I ...
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### Probability that a random polynomial has a linear factor?

What is the probability that a random degree $d$ polynomial in $\Bbb F_p[x]$ has a linear factor (root in $\Bbb F_p$) for cases $d>p$ and $d<p$? Lower bound is $\frac1p$ but definitely seems ...
### Prove the polynomial $P_a=X^5 + a$ is reducible over a field
Let $(K, +, \cdot)$ a finite field so that the polynomial $P=X^2-5$ is irreducible. Prove that: a) $1+1 \ne 0$ b) The polynomial $P_a=X^5 + a$ is reducible $\forall a \in K$ a) ...