======================= Rings and Fields ======================= Midterm: 2025-05-07T10:00-03:00 Final exam: 2025-06-25T10:00-03:00 Make-up exam: 2025-07-02T10:00-03:00 Exercises. P1 Q2.1 ======== Define characteristic of a ring. P1 Q2.2 ======== A field with characteristic zero has a subfield isomorphic to the rationals. P1 Q3.1 ======== Let :math:`C` be the ring of functions :math:`\mathbb{R}^{\mathbb{R}}` with the pointwise operations addition and product. For each :math:`r\in\mathbb{R}`, let :math:`M(r)` be the subset of :math:`C` given by :math:`M(r)=\{f\in C:f(r)=0\}`. Define the concept of maximal ideal. P1 Q3.2 ======== Show that :math:`M(r)` is a maximal ideal. P1 Q4.1 ======== Define sum and product operations between formal power series rings. P1 Q4.2 ======== Let :math:`R` be an integral domain. Prove that the ring of formal power series :math:`R[\![x]\!]` is an integral domain. `[MathSE link] `_. P1 Q5.1 ======= Define Euclidean domain. P1 Q5.2 ======= Let :math:`R` be a Euclidean domain equipped with the Euclidean function :math:`\nu :R\setminus \{0\}\rightarrow\mathbb{N}` and :math:`a\in R` nonzero. Let :math:`m` be the least integer taken by :math:`\nu`. Prove that :math:`a\in R\setminus \{0\}` is invertible iff :math:`\nu (a)=m`. `[MathSE link] `_. P2 Q1.3 E10.7 ============= Let :math:`F/E` be a field extension, and :math:`u\in F`. Prove that, if the degree of :math:`u` over :math:`E` is finite and odd, then :math:`E(u)=E(u^2)`. `[MathSE link] `_. P2 Q2.1 ======= Eisenstein's irreducibility criterion. P2 Q2.4 ======= Let :math:`p\in\mathbb{Z}` be a positive prime. For which integers :math:`n\geq 1` is the real number :math:`\sqrt[n]{p}` constructible? `Algebraic properties of Constructible numbers [wikipedia] `_. P2 Q3.2 ======= Let :math:`R` be a UFD and :math:`a_0,...,a_n\in R` not all of which are zero. Prove that :math:`1\in GCD(a_0,...,a_n)` iff for all :math:`p\in R` irreducible, there exists :math:`i\in\{0,...,n\}` such that :math:`p\nmid a_i`. Hint: For the forward direction, use contraposition. For the reverse direction, let :math:`d` be a common divisor of :math:`a_0,...,a_n`, then :math:`d=u\cdot p_1^{e_1}\cdot p_2^{e_2}\dotsb p_k^{e_k}`. P2 Q4.1 ======= Define algebraic element over smaller field. Define algebraic extension. P2 Q4.2 ======= If :math:`F/E` is an algebraic field extension and :math:`D` is a subring of :math:`F` containing :math:`E`, then :math:`D` is a field. `Subring of an algebraic field extension is a subfield [MathSE] `_. P2 Q4.3 ======= For the previous item, provide an example that shows the necessity of the algebraicness hypothesis. Hint: Consider the field extension :math:`\mathbb{Q}(\pi ,\sqrt{2})`, and the subring :math:`\mathbb{Q}[\pi ,\sqrt{2}]=\{a+b\sqrt{2} +c\pi +d\pi\sqrt{2} :a,b,c,d\in\mathbb{Q} \}` P2 Q4.4 ======= Let :math:`L` be a subfield of :math:`\mathbb{R}` and :math:`a\in\mathbb{R}` non-negative. Prove that :math:`a` is algebraic over :math:`L` iff :math:`\sqrt{a}` is algebraic over :math:`L`. `[MathSE link] `_. PSub Q1.2 ========= Let :math:`A` be a ring and :math:`S` a non-empty collection of subrings of :math:`A` such that for all :math:`B,C\in S`, there exists :math:`D\in S` such that :math:`B\cup C \subset D`. Prove that :math:`\bigcup_{B\in S}B` is a subring of :math:`A`. Hint: Apply the `subring test `_. `The union of two subrings is a subring if and only if either of the subring is contained in the other [MathSE] `_. PSub Q2.2 ========= `[MathSE link] `_. `Isomorphisms of quotient rings [MathSE] `_. PSub Q3.1 ========= Let :math:`(A_i)` be a family of rings. Define the product of rings :math:`\prod_{i\in I} A_i`. PSub Q3.3 ========= Prove or give a counterexample. The product of two integral domains is an integral domain. E5.1 PSub Q3.4 ============== List all elements of :math:`\mathbb{Z}_{12}` that are zero divisors. E9.6 PSub Q3.2 ============== Consider :math:`R=\mathbb{Z}_4^{\mathbb{N}}` with the structure of a product of rings. Let :math:`a\in R` be given by :math:`a(n)=2` for all :math:`n\in \mathbb{N}`. Prove that the polynomial :math:`ax\in R[x]` has infinite roots. PSub Q4.1 ========= Define finitely generated field extension. E10.5 PSub Q4.2 =============== Give an example of a finitely generated field extension that doesn't have finite degree. Explain. `Field extension that is finitely generated but not finite dimensional [MathSE] `_. E10.8. Same as E10.9. E10.9. Let :math:`F/E` be an algebraic extension, and :math:`D` a subring of :math:`F` containing :math:`E`. Prove that :math:`D` is a field. Provide an example that shows that the algebraicness condition is necessary. `Link 0 `_. E10.10 PSub Q4.3, Q4.4 ===================== Let :math:`F/E` be a field extension, and :math:`u,v \in F` elements of degree :math:`m,n < \infty` over :math:`E`, respectively. Prove that :math:`[E(u,v):E]\leq mn`. Prove also that, if :math:`m,n` are coprime, then :math:`[E(u,v):E]=mn`. `Extension fields of coprime degrees [MathSE] `_. E10.11. Prove that the subfields :math:`\mathbb{Q} (i)` , :math:`\mathbb{Q} (\sqrt{2})` of :math:`\mathbb{C}` are isomorphic as :math:`\mathbb{Q}`-vector spaces, but not isomorphic as fields. `Fields isomorphic as vector spaces but not as fields [MathSE] `_. Is :math:`\mathbb{Q}\sqrt{2} \cong \mathbb{Q}\sqrt{3} ?` `[MathSE Link] `_. Why :math:`\mathbb{Q}\sqrt{2}` and :math:`\mathbb{Q}\sqrt{3}` are isomorphic as :math:`\mathbb{Q}`-vector spaces but not as fields? `[MathSE link] `_. `Examples of fields that are isomorphic, infinite, and not equal `_. Comment: Page 241 of Hungerford, according to the MathSE post ":math:`\mathbb{C}` and :math:`\mathbb{Q}` are isomorphic as vector spaces but not as fields" `[Link] `_ (problem statement has a typo). E10.19 PSub Q4.5 ================ If :math:`F` is a finite field, then the cardinality of :math:`F` is :math:`p^n` for some prime :math:`p` and some :math:`n\in\mathbb{N}`. `Finite field always has prime power order [MathSE] `_.