Proving the $(1)$ is a prime element in the set T
I am wondering if there are any logical leaps and anyway for the proof to be concise.
Background information:
$T=left{left[begin{array}{cc}
a & b \
c & d
end{array}right]:, a,b,c,din mathbb{Z},, a=b=c=d right }$
and the notation for an element in $T$:
$(x)=left[begin{array}{cc}
x & x \
x & x
end{array}right], forall xin mathbb{Z}$
$(x)(y)=(2xy)$
The definition of being composite and prime in $T$:
An element $(x)$ is $textbf{composite}$ in $T$ if and only if $(x)=(y)(z)$ where $(y),(z) in T$. If $(x)$ is not composite, it is said to be $textbf{prime}$.
Proof:
Suppose $(1)$ is a composite. Then by the above Definition: $(1)=(2xy)=(x)(y)$ where $x,yin mathbb{Z}$ and $yneq0$. Thus, $1=2xy$. $x=frac{1}{2y}$. Then, $xin mathbb{Q}$ because $y in mathbb{Z}$. But $x$ is defined in $mathbb{Z}$. Therefore, $x notin mathbb{Z} implies (x)notin T$. Therefore, $(1)$ is a prime element in $T$.
Citation: Jacobson, Bernard. “Matrix Number Theory: An Example of Nonunique Factorization.” The American Mathematical Monthly, vol. 72, no. 4, 1965, pp. 399–402. JSTOR, www.jstor.org/stable/2313504.
Thanks,
Cba
matrices
asked Dec 9 '18 at 23:30
cba
13
add a comment |
I am wondering if there are any logical leaps and anyway for the proof to be concise.
Background information:
$T=left{left[begin{array}{cc}
a & b \
c & d
end{array}right]:, a,b,c,din mathbb{Z},, a=b=c=d right }$
and the notation for an element in $T$:
$(x)=left[begin{array}{cc}
x & x \
x & x
end{array}right], forall xin mathbb{Z}$
$(x)(y)=(2xy)$
The definition of being composite and prime in $T$:
An element $(x)$ is $textbf{composite}$ in $T$ if and only if $(x)=(y)(z)$ where $(y),(z) in T$. If $(x)$ is not composite, it is said to be $textbf{prime}$.
Proof:
Suppose $(1)$ is a composite. Then by the above Definition: $(1)=(2xy)=(x)(y)$ where $x,yin mathbb{Z}$ and $yneq0$. Thus, $1=2xy$. $x=frac{1}{2y}$. Then, $xin mathbb{Q}$ because $y in mathbb{Z}$. But $x$ is defined in $mathbb{Z}$. Therefore, $x notin mathbb{Z} implies (x)notin T$. Therefore, $(1)$ is a prime element in $T$.
Citation: Jacobson, Bernard. “Matrix Number Theory: An Example of Nonunique Factorization.” The American Mathematical Monthly, vol. 72, no. 4, 1965, pp. 399–402. JSTOR, www.jstor.org/stable/2313504.
Thanks,
Cba
matrices
asked Dec 9 '18 at 23:30
cba
13
add a comment |
I am wondering if there are any logical leaps and anyway for the proof to be concise.
Background information:
$T=left{left[begin{array}{cc}
a & b \
c & d
end{array}right]:, a,b,c,din mathbb{Z},, a=b=c=d right }$
and the notation for an element in $T$:
$(x)=left[begin{array}{cc}
x & x \
x & x
end{array}right], forall xin mathbb{Z}$
$(x)(y)=(2xy)$
The definition of being composite and prime in $T$:
An element $(x)$ is $textbf{composite}$ in $T$ if and only if $(x)=(y)(z)$ where $(y),(z) in T$. If $(x)$ is not composite, it is said to be $textbf{prime}$.
Proof:
Suppose $(1)$ is a composite. Then by the above Definition: $(1)=(2xy)=(x)(y)$ where $x,yin mathbb{Z}$ and $yneq0$. Thus, $1=2xy$. $x=frac{1}{2y}$. Then, $xin mathbb{Q}$ because $y in mathbb{Z}$. But $x$ is defined in $mathbb{Z}$. Therefore, $x notin mathbb{Z} implies (x)notin T$. Therefore, $(1)$ is a prime element in $T$.
Citation: Jacobson, Bernard. “Matrix Number Theory: An Example of Nonunique Factorization.” The American Mathematical Monthly, vol. 72, no. 4, 1965, pp. 399–402. JSTOR, www.jstor.org/stable/2313504.
Thanks,
Cba
matrices
asked Dec 9 '18 at 23:30
cba
13
I am wondering if there are any logical leaps and anyway for the proof to be concise.
Background information:
$T=left{left[begin{array}{cc}
a & b \
c & d
end{array}right]:, a,b,c,din mathbb{Z},, a=b=c=d right }$
and the notation for an element in $T$:
$(x)=left[begin{array}{cc}
x & x \
x & x
end{array}right], forall xin mathbb{Z}$
$(x)(y)=(2xy)$
The definition of being composite and prime in $T$:
An element $(x)$ is $textbf{composite}$ in $T$ if and only if $(x)=(y)(z)$ where $(y),(z) in T$. If $(x)$ is not composite, it is said to be $textbf{prime}$.
Proof:
Suppose $(1)$ is a composite. Then by the above Definition: $(1)=(2xy)=(x)(y)$ where $x,yin mathbb{Z}$ and $yneq0$. Thus, $1=2xy$. $x=frac{1}{2y}$. Then, $xin mathbb{Q}$ because $y in mathbb{Z}$. But $x$ is defined in $mathbb{Z}$. Therefore, $x notin mathbb{Z} implies (x)notin T$. Therefore, $(1)$ is a prime element in $T$.
Citation: Jacobson, Bernard. “Matrix Number Theory: An Example of Nonunique Factorization.” The American Mathematical Monthly, vol. 72, no. 4, 1965, pp. 399–402. JSTOR, www.jstor.org/stable/2313504.
Thanks,
Cba
matrices
matrices
asked Dec 9 '18 at 23:30
cba
13
asked Dec 9 '18 at 23:30
cba
13
edited Dec 9 '18 at 23:42
asked Dec 9 '18 at 23:30
cba
13
asked Dec 9 '18 at 23:30
cba
13
asked Dec 9 '18 at 23:30
cba
13
13
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