MATH 415 Lecture 22

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Irreducibility

Example

$f(x)=x^{4}-2x^{2}+8x+1\in \mathbb {Q} [x]$

If $f(x)=(x-a)\,g(x)$, where $\deg {g}=3$, we must have $a\mid 1$, so $a=\pm 1$. However, $f(\pm 1)\neq 0$

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = (x^2 + ax + b) \, (x^2 + cx + d)} , then we must have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = x^4 + (a+c)x^3 + (b+d+ac)x^2 + (ad+bc)x + bd} . In particular,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{cases} bd = 1 ad+bc = 8 b+d+ac = -2 a + c = 0 \end{cases}}

However, this system is inconsistent.

Theorem 23.5: Eisenstein Criterion

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p \in \mathbb{Z}} be a prime. Suppose that the polynomial Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = a_n \, x^n + \dots + a_1 \, x + a_0} is in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Z}[x]} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n \not\equiv 0 \pmod{p}} , but $a_{i}\equiv 0{\pmod {p}}$ for all $i<n$, with $a_{0}\not \equiv 0{\pmod {p^{2}}}$. Then $f(x)$ is irreducible over rationals.

Proof. Assume $f(x)=g(x)\,h(x)$ with $\deg {g},\deg {h}\geq 1$ and $g(x),h(x)\in \mathbb {Q} [x]$. Let $\deg {g}=r$ and $\deg {h}=s$ with coefficients $b_{r}\neq 0$ in $g(x)$ and $c_{s}\neq 0$ in $h(x)$:

${\begin{aligned}g(x)&=b_{r}\,x^{r}+\dots +b_{0}\\h(x)&=c_{r}\,x^{r}+\dots +c_{0}\end{aligned}}$

If $a_{0}\not \equiv 0{\pmod {p^{2}}}$, then neither $b_{0}$ nor $c_{0}$ are congruent to 0 (mod $p$). We must have $a_{0}=b_{0}\cdot c_{0}$ and $a_{n}=b_{r}\cdot c_{s}$.

Let $m$ be the smallest $k$ such that $c_{k}\not \equiv 0{\pmod {p}}$. We have

$a_{m}=b_{0}\,c_{m}+b_{1}\,c_{m-1}+\dots +{\begin{cases}b_{m}\,c_{0}&r\geq m\\b_{r}\,\underbrace {c_{m-r}} _{\equiv 0{\pmod {p}}}&r<m\end{cases}}$

This implies $a_{m}\not \equiv 0{\pmod {p}}$, so $m=n$.

$s=n$ implies $r=0$, which means $g(x)$ is a constant function. Contradiction!

quod erat demonstrandum

Example 1

Show that $x^{2}-2$ is irreducible in $\mathbb {Q} [x]$.

1. ${\sqrt {2}}\not \in \mathbb {Q}$
2. $x^{2}-2=(x-a)(x-b)$ for $a,b\in \mathbb {Z}$, then $a=\pm 1,\pm 2$
3. $p=2$, $p\nmid 1$, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p \mid 0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p \mid (-2)}
   • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 \not\equiv 0 \pmod{2}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \equiv 0 \pmod{2}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 \equiv 0 \pmod{2}} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2 \not\equiv 0 \pmod{2^2}}

Therefore this function is irreducible by Eisenstein Criterion.

Example 2

Show that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 25x^5 - 9x^4 - 3x^2 - 12} is irreducible over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Q}} .

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p = 3} .
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p^2 \nmid 12} .

Irreducible

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_p(x) = \frac{x^p-1}{x-1} = x^{p-1} + x^{p-2} + \dots + x + 1} . This is called the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} th cyclotonic polynomial.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_p(x)} is irreducible over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Q}} .

To prove this, we will need a well-defined homomorphism Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi_{x + 1} : \mathbb{Q}[x] \to \mathbb{Q}[x]} (Take a look at the Remark after 22.5).

Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi} maps Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{Q}[x]} to itself, it is an automorphism.

Proof. Seeking a contradiction, assume Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_p(x) = g(x) \, h(x)} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \deg{g}, \deg{h} \ge 1} . Because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi_{x+1}} is a homomorphism, we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi_{x+1}(\phi_p(x)) = \varphi_{x+1}(g(x) \, h(x)) = \varphi_{x+1}(g(x)) \, \varphi_{x+1}(h(x))}

Hence

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \tilde{\phi}_p(x) = \phi_p(x+1) &= g(x+1) \, h(x+1) = \tilde{g}(x) \, \tilde{h}(x) \\ &= \frac{\left( x+1 \right)^p-1}{(x+1)-1} = \frac{x^p + \binom{p}{1} + \dots + \binom{p}{i} \, x^{p-i} + \dots + \binom{p}{p} \, x + 1 - 1}{x} \end{align}}

Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} is prime, we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \binom{p}{i} \equiv 0 \pmod{p}} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i \not\equiv 0 \pmod{p}} . Thus Eisenstein Criteroin applies, so the original function is irreducible.

Section 24: Noncommutative Rings

Rings of Endomorphisms

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (A,+)} be an abelian group. A homomorphism $f:A\to A$ is called an endomorphism.

Note: An automorphism is an isomorphisom of a group onto itself, but an endomorphism is a homomorphism of a group onto itself.

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{End}(A)} be the set of endomorphisms of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} .

We define the addition operation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\varphi + \psi)(a) = \varphi(a) + \psi(a)} , and we get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle \mathrm{End}(A), + \right\rangle} is an abelian group with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma : A \to A} given by the trivial homomorphism Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma(a) = 0} .

We construct the product Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\varphi \circ \psi)(a) = \phi(\psi(a))} . This satisfies the left and right distributive laws, so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\langle \mathrm{End}(A), +, \circ \right\rangle} is a ring.