# Tagged Questions

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### order of $x \operatorname{mod} p(x)$ in $\mathbb Z_2$

I am writing a software that analyze the behavior of an LFSR given its feedback polynomial. At some point, I need to compute the order of $x \operatorname{mod} p(x)$ in $\mathbb Z_2$. In mathematical ...
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### a square in a finite field of odd order

GF(q) is a finite field of order q, where q is odd. Prove that $a\in GF(q), a\neq0$ has a root in $GF(q)$ iff $a^{(q-1)/2}=1$. I tried to prove it this way: Suppose a has a root in ...
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### The existence of primitive algebraic units modulo 3

Consider the problem of computing $$\sqrt{2} \mod 3$$ Whereas we seek a number $n$ such that $n^2 \equiv 2 \mod 3$ and furthermore it is known that both $n$ and $2n$ will satisfy this property, ...
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### Why doesn't SAGE understand reduced expressions mod p in a finite field extension?

Suppose I have a finite field $\mathbb{F}_{13}$ and I would like to adjoin an element, $\zeta$, with order $3$, since $\mathbb{F}_{13}$ does not contain one. So consider $\mathbb{F}_{13}(\zeta)$. Then ...
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### Existence of a map $\phi:\mathbb{Z}_{N^2}^* \mapsto \mathbb{F}$

Is there a map between the group of $\mathbb{Z}_{N^2}^*$ where $N$ is a composite number , a product of two equal size secure prime numbers $p$ and $q$ and a finite field $\mathbb{F}$, such that for ...
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### A hint on why if $c$ is not a square in $\mathbf{F}_p$, then $c^{(p - 1)/2} \equiv -1 \mod p$

Let $\mathbf{F}_p$ be a finite field and let $c \in (\mathbf{Z}/p)^\times$. If $x^2 = c$ does not have a solution in $\mathbf{F}_p$, then $c^\frac{p - 1}{2} \equiv -1 \mod p$. I will try to prove the ...
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### Quadratic Polynomials over $\mathbb F_p$

I'd like to know a reason why (irreducible) quadratic polynomials over $\mathbb F_p$ do not reach all numbers in $\mathbb F_p$. Example: $f(x)=x^2+3x+1$ in $\mathbb F_7$ is irreducible, i.e has no ...
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### Let $L = \mathbb F_2[X]/\langle X^4 + X + 1 \rangle$ is a field. Show $L^* = L / \{0\} = \langle X \rangle$ is cyclic.

Let $L = \mathbb F_2[X]/\langle X^4 + X + 1 \rangle$ is a field. Show $L^* = L / \{0\} = \langle X \rangle$ is cyclic. I've proven that $X^4 + X + 1$ is irreducible, so $L$ is a field. I also know ...
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### Show that the field $L = \mathbb F_3[X]/\langle X^5 - X + 1\rangle$ consists of $243$ elements and that $[X][f] = [1]$ for some $[f] \in L$.

Show that the field $L = \mathbb F_3[X]/\langle X^5 - X + 1\rangle$ consists of $243$ elements and that $[X][f] = [1]$ for some $[f] \in L$. I know that $L$ is a ring, so it is also a group with ...
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### Algorithm to find solution to $ax^2 + by^2 = 1$ in a finite field

Let $\mathbb{F}$ be a finite field, and let $a,b \in \mathbb{F}$ be given, subject to $a\ne 0, b \ne 0$. Consider the equation $$ax^2 + by^2 = 1.$$ It is guaranteed that there exists a solution to ...
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