Jacoboson 4.3 Work Problem

3.1 Show that the dimensionality of a splitting field \(E/F\) of \(f(x)\) of degree \(n\) is at most \(n!\)
solution:
Let \(x_1,x_2,\dots,x_n\) be the roots of \(f(x)\).
It is easy to see that
\([F(x_1):F]\leq n\)
because \(x_1\) is a root of \(f(x)\).
We show that for any positive integers \(i\) from 2 to \(n\), we have
\([F(x_1,\dots,x_i):F(x_1,\dots,x_{i-1})]\leq n-i+1\).
This is because the polynomial \(f(x)\) have \(i-1\) roots (\(x_1,x_2,\dots x_{i-1}\)) in \(F(x_1,\dots,x_{i-1})\), so it is of degree \(n-i-1\) over \(F(x_1,\dots,x_{i-1})\).
Finally, we have
\(\prod_{i=1}^{n}[F(x_1,\dots,x_i):F(x_1,\dots,x_{i-1})]\leq n!\).
3.2 Construct a splitting field over \(\mathbb{Q}\) of \(x^5-2\). Find its dimensionality over \(\mathbb{Q}\).
solution:
Let \(z=e^{\frac{2\pi i}{5}}\), then polynomial \(x^5-2\) can be factorized into
\((x-\sqrt[5]{2} z)(x- \sqrt[5]{2} z^2)(x- \sqrt[5]{2} z^3)(x-\sqrt[5]{2} z^4)(x-\sqrt[5]{2})\)( which is irreducible over \(\mathbb{Q}\) by Eisenstein's criterion).
We know that
\([\mathbb{Q}(\sqrt[5]{2}):\mathbb{Q}]=5\) because \(x^5-2\) is \(\sqrt[5]{2}\)'s minimal polynomial, and
\([\mathbb{Q}(\sqrt[5]{2})(z):\mathbb{Q}(\sqrt[5]{2})]=4\) because \(x^4+x^3+x^2+x+1\) is \(z\)'s minimal polynomial.
Thus, the dimensionality of splitting field \(\mathbb{Q}(\sqrt[5]{2})(z)\) is 20.
3.3 Determin a splitting field over \(\mathbb{Z}/(p)\) of \(x^{p^e}-1,e\in\mathbb{N}\).
solution:
We have \(x^{p^e}-1=(x-1)^{p^e}\). So the splitting field is \(\mathbb{Z}_p\)
3.4 Let \(E/F\) be a splitting field over \(F\) of \(f(x)\) and let \(K\) be a subfield of \(E/F\). Show that any monomorphism of \(K/F\) into \(E/F\) can be extended to an automorphism of \(E\).
solution: Let \(\eta\) be a monomorphism of \(K/F\) into \(E/F\) and let \(r_1,r_2,\dots,r_n\) be the roots of \(f(x)\) in \(E\). We have
\(E=K(r_1,r_2,\dots,r_n)\).
Let \(g(x)\) denote the minimal polynomial of \(r_1\) over \(K\), \(\bar{g}(x)\) denote the image of polynomial \(g(x)\) under \(\eta\). Obviously, This polynomial is the minimal polynomial of \(r_1\) over \(\bar{K}\) and \(r_1\) is in \(E\). Thus \(\eta\) can be extended to a monomorphism of \(K(r_1)/F\) into \(E/F\). By induction, \(\eta\) could be extended to an automorphism of \(E\).
Remark: 这里主要用了这样一个引理：
Lemma: Let \(\eta\) be an isomorphism of a field \(F\) onto a field \(\bar{F}\), and let \(E\) and \(\bar{E}\) be extension fields of \(F\) and \(\bar{F}\) respectively. Suppose \(r\in E\) is algebraic over \(F\) with minimum polynomial \(g(x)\). Then \(\eta\) can be extended to a monomorphism \(\mu\) of \(F(r)\) into \(\bar{E}\) if and only if \(\bar{g}(x)\) has a root in \(\bar{E}\), in which case the number of such extensions is the same as number of distinct roots of \(\bar{g}(x)\) in \(E\).
pf: see Basic Algebra 1, 4.3, p227.
我们这道题中, \(F\)就是\(K\), \(\bar{F}\) 就是\(K\)在\(\eta\)下的像. 我们主要验证\(\bar{g}(x)\) has a root in \(\bar{E}\) 即可. 事实上这还有一个\(k\)-代数的推广版本，大部分\(k\)-代数教材都有所以自行查看。
3.5 Let \(E\) be an extension field of \(F\) such that \([E:F]=n<\infty\). Let \(K\) be any extension field of \(F\). Use the method of the proof of Theorem 4.4 to show that the number of monomorphism of \(E/F\) into \(K/F\) does not exceed \(n\).
solution:
Similarly, we do induction on \(n\).
When \(n=1\), then the only monomorphism is identity map.
Now assume that it holds for \(n>1\).
Take \(r\in E\backslash F\). Let \(f(x)\) be the minimum polynomial of \(r\) over \(F\), where \(deg(f)=l\). Then the number of monomorphism of \(F(r)/F\) into \(K/F\) does not exceed \(l\). By induction, the number of monomorphism of \(E\) into \(K\) extending a monomorphism of \(F(r)\) into \(E\) does not exceed \(\frac{n}{l}\). Thus we finish the proof.