Explicit Field Isomorphism of Finite Fields

Problems and Solutions in Field Theory in Abstract Algebra

Problem 233

(a) Let $f_1(x)$ and $f_2(x)$ be irreducible polynomials over a finite field $\F_p$, where $p$ is a prime number. Suppose that $f_1(x)$ and $f_2(x)$ have the same degrees. Then show that fields $\F_p[x]/(f_1(x))$ and $\F_p[x]/(f_2(x))$ are isomorphic.

(b) Show that the polynomials $x^3-x+1$ and $x^3-x-1$ are both irreducible polynomials over the finite field $\F_3$.

(c) Exhibit an explicit isomorphism between the splitting fields of $x^3-x+1$ and $x^3-x-1$ over $\F_3$.

 
LoadingAdd to solve later
Sponsored Links

Proof.

(a) Fields $\F_p[x]/(f_1(x))$ and $\F_p[x]/(f_2(x))$ are isomorphic

Let $n$ be the degree of $f_1$ and $f_2$.
Since $f_1$ is irreducible over $\F_p$, the quotient field $\F_p[x]/(f_1(x))$ is the finite field of $p^n$ elements.
Similarly, so is $\F_p[x]/(f_2(x))$.

Since a finite field of $p^n$ elements are unique up to isomorphism, these two quotient fields are isomorphic.


Here, we give an explicit isomorphism. The polynomial $f_1(x)$ splits completely in the field $F_{p^n}\cong \F_p[x]/(f_2(x))$, so let $\theta$ be a root of $f_1(x)$ in $\F_p[x]/(f_2(x))$. (Note that $\theta$ is a polynomial.)
Define a map
\[\Phi: \F_p[x] \to \F_p[x]/(f_2(x))\] sending $g(x)\in \F_p[x]$ to $g(\theta)$. The map $\Phi$ is a ring homomorphism.
We want to show that the kernel $\ker(\Phi)=(f_1(x))$.

Since $\Phi(f_1(x))=f_1(\theta)=0$, we have $(f_1(x)) \subset \ker(\Phi)$.


On the other hand, if $g(x)\in \ker(\Phi)$, then we have $g(\theta)=0$.
Since $f_1(x)$ is the minimal polynomial of $\theta$, it follows that $f_1$ divides $g(x)$, and hence $g(x) \in (f_1(x))$.
Therefore we proved $\ker(\Phi)=(f_1(x))$.

By the first isomorphism theorem, we obtain an isomorphism
\[\tilde\Phi: \F_p[x]/(f_1(x)) \xrightarrow{\cong} \F_p[x]/(f_2(x)),\] where $\tilde \Phi$ maps $x$ to $\theta$.

(b) The polynomials $x^3-x+1$ and $x^3-x-1$ are irreducible over $\F_3$

Since these polynomial are of degree $3$, if they are reducible, then it has a root in $\F_3$. Evaluating these polynomials at $x=0,1,2$ shows that they have no roots in $\F_3$. Thus these two polynomial are irreducible over $\F_3$.

(c) Explicit isomorphism between the splitting fields of $x^3-x+1$ and $x^3-x-1$ over $\F_3$

By part (a), the splitting fields
\[ \F_3[x]/(x^3-x+1) \text{ and } \F_3[x]/(x^3-x-1)\] are isomorphic. In the proof of part (a), we gave an explicit isomorphism.
That is, if $\theta$ is a root of $x^3-x+1$ in the field $\F_3[x]/(x^3-x-1)$, then the map sending $x\in \F_3[x]/(x^3-x+1)$ to $\theta \in \F_3[x]/(x^3-x-1)$ gives an isomorphism.

So we want to find a root $\theta$ of $f_1(x):=x^3-x+1$.
Let $\theta=a+bx+cx^2\in \F_3[x]/(x^3-x-1)$.


Then we have
\begin{align*}
&f_1(\theta)=f_1(a+bx+cx^2)\\
&=(a+bx+cx^2)^3-(a+bx+cx^2)+1\\
&=a+bx^3+cx^6-(a+bx+cx^2)+1\\
& \text{(Note that $a^3=a$ in $\F_3$ and similarly for $b$ and $c$.)}\\
&=a+b(x+1)+c(x^2+2x+1)-(a+bx+cx^2)+1\\
&\text{(Note that $x^3=x+1$ in $\F_3[x]/(x^3-x-1)$, and thus $x^6=x^2+2x+1$.)}\\
&=2cx+b+c+1\stackrel{\text{set}}{=}0.
\end{align*}


From this we deduce that $c=0$, $b=2$ gives a root $\theta$.
For example, choosing $a=0$, we have a root $\theta=2x$ of $f_1(x)$.
Therefore the explicit isomorphism is
\[ \Phi:\F_3[x]/(x^3-x+1) \xrightarrow{\cong} \F_3[x]/(x^3-x-1),\] which sends $x$ to $\theta=2x$.


LoadingAdd to solve later

Sponsored Links

More from my site

  • Each Element in a Finite Field is the Sum of Two SquaresEach Element in a Finite Field is the Sum of Two Squares Let $F$ be a finite field. Prove that each element in the field $F$ is the sum of two squares in $F$. Proof. Let $x$ be an element in $F$. We want to show that there exists $a, b\in F$ such that \[x=a^2+b^2.\] Since $F$ is a finite field, the characteristic $p$ of the field […]
  • Characteristic of an Integral Domain is 0 or a Prime NumberCharacteristic of an Integral Domain is 0 or a Prime Number Let $R$ be a commutative ring with $1$. Show that if $R$ is an integral domain, then the characteristic of $R$ is either $0$ or a prime number $p$.   Definition of the characteristic of a ring. The characteristic of a commutative ring $R$ with $1$ is defined as […]
  • A Maximal Ideal in the Ring of Continuous Functions and a Quotient RingA Maximal Ideal in the Ring of Continuous Functions and a Quotient Ring Let $R$ be the ring of all continuous functions on the interval $[0, 2]$. Let $I$ be the subset of $R$ defined by \[I:=\{ f(x) \in R \mid f(1)=0\}.\] Then prove that $I$ is an ideal of the ring $R$. Moreover, show that $I$ is maximal and determine […]
  • $(x^3-y^2)$ is a Prime Ideal in the Ring $R[x, y]$, $R$ is an Integral Domain.$(x^3-y^2)$ is a Prime Ideal in the Ring $R[x, y]$, $R$ is an Integral Domain. Let $R$ be an integral domain. Then prove that the ideal $(x^3-y^2)$ is a prime ideal in the ring $R[x, y]$.   Proof. Consider the ring $R[t]$, where $t$ is a variable. Since $R$ is an integral domain, so is $R[t]$. Define the function $\Psi:R[x,y] \to R[t]$ sending […]
  • Prove that $\F_3[x]/(x^2+1)$ is a Field and Find the Inverse ElementsProve that $\F_3[x]/(x^2+1)$ is a Field and Find the Inverse Elements Let $\F_3=\Zmod{3}$ be the finite field of order $3$. Consider the ring $\F_3[x]$ of polynomial over $\F_3$ and its ideal $I=(x^2+1)$ generated by $x^2+1\in \F_3[x]$. (a) Prove that the quotient ring $\F_3[x]/(x^2+1)$ is a field. How many elements does the field have? (b) […]
  • The Polynomial $x^p-2$ is Irreducible Over the Cyclotomic Field of $p$-th Root of UnityThe Polynomial $x^p-2$ is Irreducible Over the Cyclotomic Field of $p$-th Root of Unity Prove that the polynomial $x^p-2$ for a prime number $p$ is irreducible over the field $\Q(\zeta_p)$, where $\zeta_p$ is a primitive $p$th root of unity.   Hint. Consider the field extension $\Q(\sqrt[p]{2}, \zeta)$, where $\zeta$ is a primitive $p$-th root of […]
  • Polynomial $x^p-x+a$ is Irreducible and Separable Over a Finite FieldPolynomial $x^p-x+a$ is Irreducible and Separable Over a Finite Field Let $p\in \Z$ be a prime number and let $\F_p$ be the field of $p$ elements. For any nonzero element $a\in \F_p$, prove that the polynomial \[f(x)=x^p-x+a\] is irreducible and separable over $F_p$. (Dummit and Foote "Abstract Algebra" Section 13.5 Exercise #5 on […]
  • Galois Group of the Polynomial  $x^p-2$.Galois Group of the Polynomial $x^p-2$. Let $p \in \Z$ be a prime number. Then describe the elements of the Galois group of the polynomial $x^p-2$.   Solution. The roots of the polynomial $x^p-2$ are \[ \sqrt[p]{2}\zeta^k, k=0,1, \dots, p-1\] where $\sqrt[p]{2}$ is a real $p$-th root of $2$ and $\zeta$ […]

You may also like...

Leave a Reply

Your email address will not be published. Required fields are marked *

More in Field Theory
Galois theory problem and solution
Galois Extension $\Q(\sqrt{2+\sqrt{2}})$ of Degree 4 with Cyclic Group

Show that $\Q(\sqrt{2+\sqrt{2}})$ is a cyclic quartic field, that is, it is a Galois extension of degree $4$ with cyclic...

Close