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In my algebra course I constantly have to do with structures that are written like this, for example: $\mathbb{F_7[x]/\langle x^3-2\rangle}$ , $\mathbb{Z_2[x]/\langle x^2+x+1\rangle}$ etc. I have to constantly prove things about them, say if they are a field and so on.

I have a big problem with this, because I don't fully grasp what they are in order to then practically use them. And most importantly; i don't know what their elements are.

I halfway understand the concept of quotients, especially in quotient groups. For example I know that $G/H=\Bbb{Z/4Z=n+4Z}$ and we have the aquivalence classes $0,1,2,3$. Even though, also here, I am a bit uncertain when we add elements in $H$ and when we multiply with elements in $H$.

But then when it gets to polynomials, I get completely lost.

Would anyone have any simple explanation(this is my first ever algebra course) in order for me to understand them and to be able to use them in the future? What do you think of when you see $\mathbb{F_7[x]/\langle x^3-2\rangle}$ and have to prove for example that it is the field. How can I better picture it in my head?

Thanks to everyone in advance

katrin
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1 Answers1

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This is a congruence class of a polynomial ring, modulo the principal ideal generated by a polynomial.

As a standard example, the quotient $\mathbf R[X]/(X^2+1)$ is a possible definition for the set of complex numbers. As polynomials with coefficients in $\mathbf R$ are endowed with euclidean division, the congruence class of a polynomial $P(X)$ has a canonical representative, namely the remainder of the division of $P(X)$ by $X^2+1$. If we denote the congruence class of $X$ by $i$, a congruence class representative is written as $a+bi\enspace (a,b\in\mathbf R)$.

Now the quotient of a polynomial ring over a field $K[X]/\bigl(F(X)\bigr)$ is a field if & only if $F(X)$ is irreducible.

Bernard
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