§4
Generated Subfields
Let \(L/K\) be a field extension and \(S \subseteq L\). The field generated by \(S\) over \(K\), written \(K(S)\), is the smallest subfield of \(L\) containing both \(K\) and \(S\). We say \(L/K\) is generated by \(S\) if \(L = K(S)\). The ring generated by \(S\) over \(K\), written \(K[S]\), is the smallest subring of \(L\) containing \(K\) and \(S\).
Note that the field generated by \(S\) over \(K\) is the intersection of all subfields of \(L\) that contain both \(K\) and \(S\). Similarly, the ring generated by \(S\) over \(K\) is the intersection of all subrings of \(L\) that contain both \(K\) and \(S\).
An extension \(L/K\) is finitely generated if there exists a finite subset \(S \subset L\) such that \(L = K(S)\).
Let \(k\) be a field. Consider the field extension \(k(x_1,x_2,\ldots,x_n)/k\), where \(k(x_1,x_2,\ldots,x_n)\) is the field of rational functions in \(n\) variables over \(k\). Show that \(k(x_1,x_2,\ldots,x_n)\) is generated by the elements \(x_1,x_2,\ldots,x_n\) over \(k\), that is, \[ k(x_1,x_2,\ldots,x_n) = k(\{x_1,x_2,\ldots,x_n\}). \] Also show that the polynomial ring \(k[x_1,\ldots,x_n]\) is generated by \(x_1,\ldots,x_n\) over \(k\), that is, \[ k[x_1,\ldots,x_n] = k[\{x_1,\ldots,x_n\}]. \]
The field of rational functions \(k(x_1,x_2,\ldots,x_n)\) in \(n\) variables over \(k\) is the field of fractions of the integral domain \(k[x_1,\ldots,x_n]\). Moreover, \(k(x_1,x_2,\ldots,x_n)\) is generated by \(x_1,x_2,\ldots,x_n\) over \(k\), and hence the extension \(k(x_1,x_2,\ldots,x_n)/k\) is finitely generated. Furthermore, the ring \(k[x_1,\ldots,x_n]\) is generated by \(x_1,\ldots,x_n\) over \(k\).
Let \(L/K\) be a field extension, and let \(S\) be a subset of \(L\). Show that the field generated by \(S\) over \(K\) is equal to the set of all elements of \(L\) that can be expressed as a rational function in the elements of \(S\) with coefficients in \(K\). In other words, show that \[ K(S) = \bigcup_{\substack{S_f \subset S \\ |S_f| < \infty}} K(S_f). \] Also show that the ring generated by \(S\) over \(K\) is equal to the set of all elements of \(L\) that can be expressed as a polynomial in the elements of \(S\) with coefficients in \(K\). In other words, show that \[ K[S] = \bigcup_{\substack{S_f \subset S \\ |S_f| < \infty}} K[S_f]. \]
Let \(L/K\) be a field extension, and let \(\alpha_1,\alpha_2,\ldots,\alpha_n \in L\). Then the field \(K(\{\alpha_1,\alpha_2,\ldots,\alpha_n\})\) is also denoted by \(K(\alpha_1,\alpha_2,\ldots,\alpha_n)\).
Show that \(\mathbb{Q}(\sqrt{2})\) and \(\mathbb{Q}(\sqrt{3})\) are not isomorphic. What can be said about \(\mathbb{Q}(\sqrt{2},\sqrt{6})\) and \(\mathbb{Q}(\sqrt{3},\sqrt{6})\)?
Show that \(\mathbb{Q}(\sqrt{2}) = \mathbb{Q}[\{\sqrt{2}\}]\).
Show that \(\mathbb{Q}(\sqrt{2},\sqrt{3}) = \mathbb{Q}(\sqrt{2}+\sqrt{3})\).
Show that \(\mathbb{Q}(\sqrt{2},\omega) = \mathbb{Q}(\sqrt{2}+\omega)\), where \(\omega\) is a primitive cube root of unity in \(\mathbb{C}\).
Let \(n\) be a positive integer, and let \(\zeta_n\) denote a primitive \(n\)-th root of unity in \(\mathbb{C}\). Then the extension \(\mathbb{Q}(\zeta_n)\) of \(\mathbb{Q}\), generated by \(\zeta_n\), is called the cyclotomic extension of \(\mathbb{Q}\) obtained by adjoining the \(n\)-th roots of unity.
Determine the degree of \(\mathbb{Q}(\zeta_n)/\mathbb{Q}\) for \(n = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\).
§5
Degree of Field Extensions
If \(L/K\) is a field extension, then \(L\) can be viewed as a vector space over \(K\), where scalar multiplication is given by the field multiplication.
Show that the map \(\mathbb{C} \times \mathbb{C} \to \mathbb{C},\ (a,b) \mapsto \bar{a}b\), where \(\bar{a}\) denotes the complex conjugate, defines a \(\mathbb{C}\)-linear structure on \(\mathbb{C}\) making it a vector space over \(\mathbb{C}\).
Let \(L/K\) be a field extension. We say \(L\) is a finite extension of \(K\) if \(L\) is finite-dimensional over \(K\). The degree is \([L:K] = \dim_K L\). If \(L\) is not finite-dimensional we say the extension is infinite.
Show that \([L:K] = 1\) if and only if \(L = K\).
Show that \([k(x):k] = \infty\) for any field \(k\).
Show that \(k(t)/k(t^2)\) is a finite extension of degree \(2\).
Determine the degrees of the extensions: \[ \mathbb{Q}(\sqrt{2})/\mathbb{Q},\quad \mathbb{Q}(\sqrt{2},\sqrt{3})/\mathbb{Q},\quad \mathbb{Q}(\sqrt{2},\sqrt[3]{3})/\mathbb{Q},\quad \mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5})/\mathbb{Q}. \]
Determine the degrees over \(\mathbb{Q}\): \(\mathbb{Q}(i),\ \mathbb{Q}(\omega),\ \mathbb{Q}(\zeta_n)\).
Let \(L/K\) be a finite extension of degree \(n\). Show that any element of \(L\) is a root of a polynomial of degree at most \(n\) with coefficients in \(K\).
Now let's look at some examples of finite extensions.
Consider \(\mathbb{Q}(\sqrt{2})/\mathbb{Q}\). Any element can be written \(a + b\sqrt{2}\), \(a,b \in \mathbb{Q}\). The set \(B = \{1, \sqrt{2}\}\) is a basis, so \([\mathbb{Q}(\sqrt{2}):\mathbb{Q}] = 2\).
Let \(L/K\), \(K/F\) be field extensions. If both are finite then \(L/F\) is finite and \[ [L:F] = [L:K][K:F]. \]
If either \([L:K]\) or \([K:F]\) is infinite then \([L:F]\) is also infinite (an infinite \(K\)-basis of \(L\) is also an infinite \(F\)-basis).
Let \(\{u_1,\ldots,u_m\}\) be a basis of \(L/K\) and \(\{v_1,\ldots,v_n\}\) a basis of \(K/F\). We claim \(B = \{u_i v_j : 1 \le i \le m,\ 1 \le j \le n\}\) is a basis of \(L/F\).
Spanning. For any \(\alpha \in L\), write \(\alpha = \sum_i c_i u_i\) with \(c_i \in K\), then write each \(c_i = \sum_j d_{ij} v_j\) with \(d_{ij} \in F\). Then \(\alpha = \sum_{i,j} d_{ij} u_i v_j\).
Independence. Suppose \(\sum_{i,j} e_{ij} u_i v_j = 0\). Grouping: \(\sum_i \bigl(\sum_j e_{ij} v_j\bigr) u_i = 0\). Since \(\{u_i\}\) is linearly independent over \(K\), each \(\sum_j e_{ij} v_j = 0\). Since \(\{v_j\}\) is linearly independent over \(F\), all \(e_{ij} = 0\). Hence \([L:F] = |B| = mn = [L:K][K:F]\). \(\square\)
If an extension has prime degree, show that it has no proper intermediate extensions.
Let \(L/K\) be a field extension and \(\alpha_1,\ldots,\alpha_n \in L\). Then \(K(\alpha_1,\ldots,\alpha_n)/K\) is finite if and only if each \(K(\alpha_i)/K\) is finite. Moreover, if finite, \[ [K(\alpha_1,\ldots,\alpha_n):K] \le \prod_{i=1}^n [K(\alpha_i):K]. \]
§6
Algebraic Elements and Algebraic Extensions
Let \(L/K\) be a field extension, \(\alpha \in L\).
- Show that multiplication-by-\(\alpha\) is a \(K\)-linear map \(L \to L\).
- If \(L/K\) is finite, its characteristic polynomial is monic of degree \([L:K]\) and vanishes at \(\alpha\).
- If \(\alpha\) is algebraic over \(K\) and \(L = K(\alpha)\), show the characteristic polynomial equals the minimal polynomial of \(\alpha\).
An element \(\alpha \in L\) is algebraic over \(K\) if there exists a nonzero \(f(x) \in K[x]\) with \(f(\alpha) = 0\). Otherwise \(\alpha\) is transcendental over \(K\).
Is every algebraic extension finitely generated? Prove or give a counterexample.
Let \(L/K\) be a field extension, \(\alpha \in L\). The following are equivalent:
- \(K(\alpha)/K\) is finite.
- \(\alpha\) is algebraic over \(K\).
Moreover, \(\alpha\) is algebraic over \(K\) iff \(\alpha^n\) is algebraic over \(K\) for some positive integer \(n\).
If \(\alpha \in L\) is algebraic over \(K\), its minimal polynomial over \(K\) is the unique monic polynomial \(f(x) \in K[x]\) of least degree with \(f(\alpha) = 0\).
Let \(L/K\) be a field extension and \(\alpha \in L\) algebraic over \(K\). Then:
- The minimal polynomial of \(\alpha\) is irreducible in \(K[x]\).
- Any \(f(x) \in K[x]\) with \(f(\alpha) = 0\) is divisible by the minimal polynomial.
- \([K(\alpha):K]\) equals the degree of the minimal polynomial.
- \(K(\alpha) = K[\alpha]\).
If \(L/K\) is a finite extension, then \(L\) is algebraic over \(K\) and finitely generated as a field extension of \(K\).
Let \(L/K\) be a field extension and \(X \subseteq L\) with every element of \(X\) algebraic over \(K\). Then \(K(X)/K\) is algebraic. Moreover, if \(X\) is finite then \(K(X)/K\) is finite.
Let \(L/K\), \(K/F\) be algebraic extensions. Then \(L/F\) is also algebraic.
Let \(L/K\) be a field extension. The set of all elements of \(L\) algebraic over \(K\) forms a subfield of \(L\) — it is the largest algebraic extension of \(K\) contained in \(L\).
The algebraic closure of \(K\) in \(L\) is the set of all elements of \(L\) algebraic over \(K\). The extension \(L/K\) is an algebraic extension if every element of \(L\) is algebraic over \(K\).
Let \(\overline{\mathbb{Q}}\) denote the algebraic closure of \(\mathbb{Q}\) in \(\mathbb{C}\). Show that \(\overline{\mathbb{Q}}\) contains an extension of \(\mathbb{Q}\) of degree \(n\) for every positive integer \(n\). Using this, show that \(\overline{\mathbb{Q}}/\mathbb{Q}\) is infinite.