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I am reading Chapter 2: Commutative Rings in Joseph Rotman's book, Advanced Modern Algebra (Second Edition).
I need help with Exercise 2.47 on page 114.
Problem 2.47 reads as follows:
View attachment 2692I need help with showing that $$f(x)$$ has a root $$ \alpha \in \mathbb{F}_4 $$.
My work on this part of the problem is as follows:
The elements of $$ \mathbb{F}_4 $$ are 0, 1, 2 and 3.
Now we proceed as follows:
$$ f(0) = 0^2 + 0 + 1 = 1_4 \ne 0_4 $$
$$ \Longrightarrow 0_4 $$ is not a root of $$f(x)$$ in $$ \mathbb{F}_4 $$
$$ f(1) = 1^2 + 1 + 1 = 3_4 \ne 0_4 $$
$$ \Longrightarrow 1_4 $$ is not a root of $$f(x)$$ in $$ \mathbb{F}_4 $$
$$ f(2) = 2^2 + 2 + 1 = 7_4 \ne 0_4 $$
$$ \Longrightarrow 2_4 $$ is not a root of $$f(x)$$ in $$ \mathbb{F}_4 $$
$$ f(3) = 3^2 + 3 + 1 = 13_4 \ne 0_4 $$
$$ \Longrightarrow 3_4 $$ is not a root of $$f(x)$$ in $$ \mathbb{F}_4 $$
... ... BUT I seem to have shown (wrongly I'm sure!) that f(x) has no root in $$ \mathbb{F}_4 $$?
Can someone please help me with this issue?
Peter
I need help with Exercise 2.47 on page 114.
Problem 2.47 reads as follows:
View attachment 2692I need help with showing that $$f(x)$$ has a root $$ \alpha \in \mathbb{F}_4 $$.
My work on this part of the problem is as follows:
The elements of $$ \mathbb{F}_4 $$ are 0, 1, 2 and 3.
Now we proceed as follows:
$$ f(0) = 0^2 + 0 + 1 = 1_4 \ne 0_4 $$
$$ \Longrightarrow 0_4 $$ is not a root of $$f(x)$$ in $$ \mathbb{F}_4 $$
$$ f(1) = 1^2 + 1 + 1 = 3_4 \ne 0_4 $$
$$ \Longrightarrow 1_4 $$ is not a root of $$f(x)$$ in $$ \mathbb{F}_4 $$
$$ f(2) = 2^2 + 2 + 1 = 7_4 \ne 0_4 $$
$$ \Longrightarrow 2_4 $$ is not a root of $$f(x)$$ in $$ \mathbb{F}_4 $$
$$ f(3) = 3^2 + 3 + 1 = 13_4 \ne 0_4 $$
$$ \Longrightarrow 3_4 $$ is not a root of $$f(x)$$ in $$ \mathbb{F}_4 $$
... ... BUT I seem to have shown (wrongly I'm sure!) that f(x) has no root in $$ \mathbb{F}_4 $$?
Can someone please help me with this issue?
Peter