Exercise 5.2.8 (Algebraic elements form subfield)

Question

Let $M:K$ be a field extension and write $L$ for the set of elements of $M$ algebraic over $K$. By imitating the proof of Proposition 5.2.7,
prove that $L$ is a subfield of $M$.

Hassaan Naeem · Accepted Answer

extbf{Solution:}
We have that $L = \{\alpha \in M: [K(\alpha):K]<\infty\}$.\
Then $\forall \alpha,\beta \in L,\ [K(\alpha,\beta):K] \leq [K(\alpha):K][K(\beta):K] < \infty$\
Now $\alpha+\beta \in K(\alpha,\beta)$, so $K(\alpha+\beta) \subseteq K(\alpha,\beta)$, hence \
$[K(\alpha+\beta):K] \leq [K(\alpha,\beta):K]<\infty$, giving $\alpha+\beta \in L$. Similarly, $\alpha \cdot \beta \in L$.\
Then $\forall \alpha \in L,\ [K(-\alpha):K] = [K(\alpha):K]<\infty$, giving $-\alpha \in L$.
Similarly, $1/\alpha \in L$ (if $\alpha 
eq 0$), and clearly $0,1 \in L \quad \square$

Exercise 5.2.8 (Algebraic elements form subfield)

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Exercise 5.2.8 (Algebraic elements form subfield)

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