# 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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### Irreducible polynomial on $\mathbb{Z}_2$-field

I've found a theorem in the book "Linear Groups" (Dickson, 1901, p.16.): "In $\mathbb{Z}_2$, the degrees of the irreducible divisors of $x^{2^m}-x$ are divisors of $m$." I've read the prove in this ...
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### Primes is in P, proof of hendrik Lenstra Jr. lemma

In the paper describing AKS primality test : http://annals.math.princeton.edu/wp-content/uploads/annals-v160-n2-p12.pdf On page no. 8 Lemma 4.7 last paragraph, I cannot understand how number of ...
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### Find the condition of $h$ satisfy $x^3+1$ divides $x^{2h+1}+1$ over $\mathbb{Z}_2$.

Find the condition of $h$ satisfy $x^3+1$ divides $x^{2h+1}+1$ over $\mathbb{Z}_2$ and find the quotient polynomial. (Sorry for my bad English)
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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 ...
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### 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) ...
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### Show that the simple extension $K(X)$ of $K$ has intermediate fields $K \varsubsetneqq F \varsubsetneqq K(X)$.

Show that the simple extension $K(X)$ of $K$ has intermediate fields $K \varsubsetneqq F \varsubsetneqq K(X)$. I have tried this: Suppose that $K \subseteq E \subseteq K(X)$. Let $\alpha \in K$ and ...
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### Is there an algorithm to find the splitting field of a polynomial over galois finite fields?

how can I find the splitting field of polynomial $x^{13}+1$ over $GF(2)$?
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### Finite separable fields extensions and discriminant

I am supposed to prove that for a finite separable field extension $L/K$, the discriminant $Discr_{L/K}$ is not zero. (For a basis $\{a_1,\ldots,a_n\}$ of $L$ over $K$, the discriminant is defined by ...
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### Splitting field of $x^9-x$ over $\mathbb{Z}_3$.

Let $F$ be a splitting field of $x^9-x$ over $\mathbb{Z}_3$. $1$. Show that there is an $\alpha \in F - \mathbb{Z}_3$ such that $\alpha^2=2$. $2$. Show that $\mathbb{Z}_3(\alpha)$ is a splitting ...
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### Finding a basis for a field

I have a polynomial f(x) = $x^3+x^2+1$ in $\mathbb{Z}_5[x]$ and it is given that F = $\mathbb{Z}_5[x]$/$<f(x)>$ = $\mathbb{Z}_5(\alpha)$ where $\alpha =x+<f(x)>$. I want to find a basis ...
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### Galois Theory.Subgroups of Galois Group

How to List the subgroups of the Galois group in general.Im not interested in a specific Example but to make it easier. Supposes the galois group $G=Gal[Q(v,i):Q]$ $$v= \sqrt[4]{2}$$ I know how to ...
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### What can we say about the dimension of $[ E : F]$ if $F finite$ and $f \in F[x]$ min. polynomial [closed]

suppose I have a field $F$ and $\alpha \notin F$. $F$ is finite so $char(F)$ is some $p \in \mathbb{N}$ when I have a minimal polynomial $f_\alpha \in F[x]$ with $deg(f_\alpha)=n$ then the dimension ...
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### Probability that a random polynomial over a finite field can be factorized to linear terms.

Suppose that $f\in\mathbb{F}_p[x]$ is a degree $d$ random univariate polynomial with coefficients from a finite field $\mathbb{F}_p$. What is the probability that $f$ can be written as: f(x)=\prod_{...
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### There is no irreducible polynomial of largest degree in $\mathbf{F}_q[x]$

I am asked to prove or disprove that given a finite field $\mathbf{F}_q$, the ring $\mathbf{F}_q[x]$ contains irreducible polynomials of arbitrarily large degree. I couldn't think of a reason why this ...
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### What is the Galois group $Gal(E/F)$ when $F=GF(5^3)$ and $E=GF(5^{24})$

What is the Galois group $Gal(E/F)$ when $F=GF(5^3)$ and $E=GF(5^{24})$. I have attempted to describe the Galois group, but I've become stuck, and it's entirely possible that I've made mistakes as ...
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### vector space homomorphism for $Map(\mathbb{F}_{5} , \mathbb{F}_{5})$

I'm currently stuck at a mathematical problem and I really don't know where to start.. Since I'm not an expert in Algebra over finite fields... It goes "Define a $\mathbb{F}_{5}$-vector space ...
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### Is the primitive element of an extension of a finite field expressible as a linear combination of adjoined elements?

Suppose $F$ is a field, and $\alpha,\beta$ are algebraic, separable elements over $F$. If $F$ is infinite, the proof of the primitive element theorem gives some $c\in F$ such that \$F(\alpha,\beta)=F(...