§1
Fields
The following are examples of fields.
- The field of rational numbers \(\mathbb{Q}\).
- The field of real numbers \(\mathbb{R}\).
- The field of complex numbers \(\mathbb{C}\).
- For any prime \(p\), the ring \(\mathbb{Z}/p\mathbb{Z}\) of integers modulo \(p\) is a field, denoted \(\mathbb{F}_p\).
- For any prime \(p\) and positive integer \(n\), there exists a unique (up to isomorphism) finite field of order \(p^n\), denoted \(\mathbb{F}_{p^n}\).
Show that the following are not fields.
- The ring of integers \(\mathbb{Z}\).
- The polynomial ring \(\mathbb{Q}[x]\) in one variable over \(\mathbb{Q}\).
- The polynomial ring \(\mathbb{R}[x,y]\) in two variables over \(\mathbb{R}\).
Show that any finite integral domain is a field.
Show that any finite field has order \(p^n\) for some prime \(p\) and positive integer \(n\).
For any field \(k\), the ring of polynomials in one variable \(x\) over \(k\) is denoted \(k[x]\). The field of rational functions \(k(x)\) consists of all elements of the form \(\dfrac{f(x)}{g(x)}\) where \(f(x), g(x) \in k[x]\) and \(g(x) \neq 0\). It is the field of fractions of the integral domain \(k[x]\).
For any field \(k\), the ring of formal power series in one variable \(x\) over \(k\) is denoted \(k[[x]]\). The field of formal Laurent series \(k((x))\) consists of generalized formal power series \[ \sum_{n \ge n_0} a_n x^n \] for some integer \(n_0\), where \(a_n \in k\) for all \(n \ge n_0\) and \(a_{n_0} \neq 0\). It is the field of fractions of \(k[[x]]\).
§2
Field Homomorphisms and Isomorphisms
Let \(K\) and \(L\) be fields. A field homomorphism \(\sigma: K \to L\) preserves the field operations: for all \(a, b \in K\),
- \(\sigma(a+b) = \sigma(a) + \sigma(b)\),
- \(\sigma(ab) = \sigma(a)\sigma(b)\),
- \(\sigma(1_K) = 1_L\).
A homomorphism \(\sigma: K \to L\) is an isomorphism if there exists a homomorphism \(\tau: L \to K\) with \(\tau \circ \sigma = \mathrm{id}_K\) and \(\sigma \circ \tau = \mathrm{id}_L\).
Show that if \(\sigma: K \to L\) is a field homomorphism, then \(\sigma\) is injective. Is \(\sigma\) necessarily surjective? If yes, prove it. If not, give a counterexample.
Let \(\sigma: K \to L\) be a field homomorphism. Show that \(\sigma\) is an isomorphism if and only if \(\sigma\) is surjective.
§3
Extension of Fields
Let \(K\) be a field. A field extension of \(K\) is a field \(L\) that contains \(K\) as a subfield. We write \(L/K\) to indicate that \(L\) is an extension of \(K\). We call \(K\) the base field and \(L\) the extension field.
- \(\mathbb{C}\) is an extension of \(\mathbb{R}\); \(\mathbb{R}\) is an extension of \(\mathbb{Q}\).
- For any field \(k\), \(k(x)/k\) and \(k((x))/k\) are field extensions.
Let \(L/K\) be a field extension. An intermediate extension is a field \(M\) with \(K \subseteq M \subseteq L\). We say \(M/K\) is a subextension of \(L/K\).
For any field \(k\), the field \(k(x^2)\) is an intermediate extension of \(k(x)/k\).
For any field \(k\), the field \(k(x)\) is an intermediate extension of \(k((x))/k\).
Let \(k(t)\) be the field of rational functions over \(k\). Show that any element of \(k(t)\) is a root of some quadratic polynomial with coefficients in \(k(t^2)\). Determine all field homomorphisms from \(k(t)\) to itself that fix \(k(t^2)\) pointwise.