# Algebra, Artin > Source: https://ollybritton.com/notes/textbooks/algebra/ · Updated: 2025-09-30 ![algebra.jpg](https://ollybritton.com/assets/attachments/img/algebra.jpg) A textbook written by [Michael Artin](https://en.wikipedia.org/wiki/Michael_Artin) (son of [Emil Artin](https://en.wikipedia.org/wiki/Emil_Artin)!) on algebra. - Relevant university courses: - Particularly helpful for: - [Course - Rings and Modules HT24](https://ollybritton.com/notes/uni/part-a/ht24/rings-and-modules/) - [Course - Galois Theory HT25](https://ollybritton.com/notes/uni/part-b/ht25/galois-theory/) - But also relevant for: - [Course - Linear Algebra MT22](https://ollybritton.com/notes/uni/prelims/mt22/linear-algebra/) - [Course - Linear Algebra II HT23](https://ollybritton.com/notes/uni/prelims/ht23/linear-algebra/) - [Course - Groups and Group Actions HT23](https://ollybritton.com/notes/uni/prelims/ht23/groups/) - [Course - Groups and Group Actions TT23](https://ollybritton.com/notes/uni/prelims/tt23/groups/) - [Course - Linear Algebra MT23](https://ollybritton.com/notes/uni/part-a/mt23/linear-algebra/) ### Chapter 15, "Fields" - **Examples of Fields** - Definition: $F \subset K$ fields, $K/F$ is a field extension - Definition: A number field is a subfield of $\mathbb C$ - Definition: Algebraic number field, a number field where all elements are algebraic over $\mathbb Q$ - Definition: Finite fields contain a finite amount of elements - Remark: Any finite field contains contains a copy of $\mathbb F_p$ and so is a field extension of $\mathbb F_p$ - Definition: Extensions of $F = \mathbb C(t)$ (rational functions over $\mathbb C$) are called function fields - Remark: Function fields can be specified implicitly in terms of an equation $f(t, x) = 0$ where $f$ is an irreducible complex polynomial in the variables of $t$ and $x$ - Example: $f(t, x) = x^2 - t^3 + t$ leads to the function field $p + q \sqrt{t^3 - t}$ where $p$ and $q$ are rational functions in $t$ - **Algebraic and transcendental elements** - Definition: If $\alpha \in K$ and $K/F$, then $\alpha$ is algebraic over $F$ if $\alpha$ is the root of a monic polynomial with coefficients in $F$. - Definition: If $\alpha$ is not algebraic over $F$, then it is transcendental - Remark: Consider the substitution homomorphism $\phi : F[x] \to K$ defined by $x \mapsto \alpha$. If $\phi$ is injective, then $\alpha$ is transcendental. Otherwise, the kernel is nonzero and $\alpha$ is algebraic. - Remark: Since $F[x]$ is a principal ideal domain, the kernel of $\phi$ is generated by a single element $f$. - Proposition: The generator for the kernel is the minimal polynomial, and the minimal polynomial has all the properties you would expect - Definition: $F(\alpha)$ is the smallest subfield of $K$ that contains $F$ and $\alpha$ - Definition: $F[\alpha]$ is the ring generated by $\alpha$ over $F$ - Remark: $F(\alpha)$ is isomorphic to the field of fractions of $F[\alpha]$. - Remark: If $\alpha$ is transcendental over $F$, then $F[x] \to F[\alpha]$ is an isomorphism and $F(\alpha)$ is isomorphic to the field $F(x)$ of rational functions. In this case, every field extension is isomorphic. - Proposition: $F[x]/\langle f \rangle \cong F[\alpha]$. - Proposition: If $\alpha_1, \ldots, \alpha_k$ are algebraic, $F[\alpha_1, \ldots, \alpha_k]$ is equal to the field $F(\alpha_1, \ldots, \alpha_k)$. - Proposition: Degree of minimal polynomial is equal to degree of field extension and powers of algebraic number give a basis - Proposition: If $\alpha$ and $\beta$ are algebraic over $F$, then $F(\alpha)$ and $F(\beta)$ are isomorphic by the map that sends $\alpha$ to $\beta$ and is the identity on $F$ iff the irreducible polynomials $\alpha$ and $\beta$ over $F$ are equal - F: Definition: Suppose $K$ and $K'$ are extensions of the same field $F$, an isomorphism $\phi : K \to K'$ that restricts to the identity on the subfield $F$ is called an $F$-isomorphism, and $K$ and $K'$ are isomorphic extension fields - Proposition: Isomorphisms of field extensions map roots to roots - **The degree of a field extension**: - Definition: Degree of a field extension - Lemma: $F/K$ has degree 1 iff $F = K$ - Lemma: Element $\alpha$ of a field extension $K$ has degree $1$ over $F$ iff $\alpha$ is an element of $F$ - Proposition: If $F$ is a field where $\text{char }F \ne 2$, you can obtain any extension $K$ by adjoining a square root, and if the field is missing a square root, adding it gives a degree 2 field extension - Remark: Not all cubic extensions can be obtained by adjoining a cube root - Proposition: $F(\alpha)$ degree of $\alpha$ over $F$ is the degree of the field extension - Proposition: $\alpha$ is algebraic iff $F(\alpha) : F$ is finite - Proposition: Tower law - Proposition: If $K/F$ is a field extension and $\alpha \in K$, if $\alpha$ is algebraic over $F$ then its degree over $F$ divides $n$. - Proposition: Degree of intermediate field can't be more than the degree of the outside field - Proposition: Every field extension generated by finitely many algebraic elements is a finite extension. A finite extension is generated by finitely many elements - Proposition: If $K/F$ is a field extension, the set of algebraic elements is a subfield - Corollary: If $K/F$ is a field extension of prime degree, then if $\alpha$ is not in $F$, the degree of the field extension is $p$ - Corollary: Let $\mathcal K$ be an extension field of a field $F$, let $K$ and $F'$ be subfields of $\mathcal K$ that are finite extensions of $F$, and let $K'$ denote the subfield of $\mathcal K$ generated by the two fields $K$ and $F'$ together. Then if $[K' : F] = N$, $[K : F] = m$, and $[F' : F] = n$. Then $m$ and $n$ divide $N$, and $N \le mn$. - Example: $[\mathbb Q (\alpha_1, \alpha_2, \alpha_3) : \mathbb Q] = 6$ where $\alpha_i$ are the roots of $x^3 - 2$. - Example: If $\alpha = \sqrt[3]{2}$ and $\beta$ is a root of $x^4 + x + 1$, since $3$ and $4$ are relatively prime, $[\mathbb Q(\alpha, \beta) : \mathbb Q] = 12$ - Example: Computing powers of $\gamma = \sqrt 2 + \sqrt 3$ and spotting a linear dependence gives you the minimal polynomial ### Chapter 16, "Galois Theory" - Symmetric functions - Definition: symmetric polynomials are the polynomials that are invariant under permutations of the variables $f(u_1, \ldots, u_n) \mapsto f(u_{1\sigma}, \ldots, u_{n\sigma})$. - Remark: The symmetric polynomials form a subring of the polynomial ring $R[u]$. - Definition: $\sigma$ defines an automorphism of $R[u]$, and since it is the identity on constant polynomials, it's called an $R$-automorphism. - Remark: $S_n$ then acts on $R[u]$ by $R$-automorphisms. - Remark: A polynomial is symmetric if the coefficients of monomials in the same orbit have the same coefficient. - Definition: Orbit sum: the sum of all polynomials in the same orbit, these form a basis of symmetric polynomials: $1$, $u_1 + u_2 + u_3$, $u^2_1 + u^2_2 + u^2_3$, $u_1 u_2 + u_1 u_3 + u_2 u_3$, and so on. - Definition: Elementary symmetric functions are special symmetric polynomials defined by - $s_1 = \sum_i u_i = u_1$ - $s_2 = \sum_{i < j} u_i u_j = u_1 u_2 + u_1 u_3 + \cdots$ - $s_3 = \sum_{i < j < k} u_i u_j u_k = u_1 u_2 u_3 + \cdots$ - ...and so on. - Remark: Elementary symmetric functions are the coefficients of the polynomial with roots $u_1, \ldots, u_n$ up to sign, e.g. $P(x) = (x - u_1)(x - u_2) = x^2 - (u_1 + u_2)x + (u_1 u_2)$. - Lemma: Can obtain the coefficients of a polynomial by evaluating the roots on the elementary symmetric functions, e.g. Vieta's formulas - Theorem: Symmetric functions theorem: Every symmetric polynomial $g(u_1, \ldots, u_n)$ with coefficients in a ring $R$ can be written in a unique way as a polynomial in the elementary functions (not the same as being a basis) - This explains some of the A-level "roots of polynomials" topic, where you'd be asked to calculate e.g. $\alpha^2 + \beta^2$ just from the coefficients $(x-\alpha)(x-\beta) = x^2 - Bx + C$. $\alpha^2 + \beta^2$ is a symmetric polynomial, but it is not an elementary symmetric function. - This result says that it's always possible to find an expression where any symmetric polynomial in the roots can be written uniquely as a polynomial in the elementary symmetric functions, so if you know the roots, you also know how to calculate any symmetric function of the roots - Examples: - $u^2_1 + \cdots + u^2_n = s_1^2 - 2s_2$ - $u_1 u_2^2 + u_2 u_1^2 + u_1 u_3^2 + u_3 u_1^2 + u_2 u_3^2 + u_3 u_2^2 = s_1 s_2 - 3 s_3$ - Corollary: Suppose $K/F$ is a splitting field for some polynomial $f \in F[x]$ with roots $\alpha_1, \ldots, \alpha_n \in K$. Then if $g$ is a symmetric polynomial, $g(\alpha_1, \ldots, \alpha_n) \in K$. - Proposition: Can build symmetric polynomials out of symmetric polynomials where we substitute the orbit of any polynomial under $S_n$ for the variables - The discriminant - Definition: The discriminant is the product of the squares of the differences of the roots, i.e. if $u_1, \ldots, u_n$ are the roots, then $D(u) = \prod_{i < j} (u_i - u_j)^2$. - Remark: $D(u)$ is a symmetric polynomial with integer coefficients - Remark: If $\alpha_1, \ldots, \alpha_n$ are elements of a field, then $D(\alpha) = 0$ iff two of the elements are equal - Remark: The symmetric function theorem says that the discriminant can always be as a polynomial of the elementary symmetric functions - Remark: For roots $u_1, u_2$, we have $D = (u_1 - u_2)^2 = s_1^2 - 4s_2$, which is the normal formula for the discriminant of a quadratic, and it's the difference between the two roots! - Remark: $D$ gets very complicated for large $n$ - Splitting fields - Definition: A splitting field for $f$ is extension field $K/F$ such that: - $f$ splits completely in $K$ - $K$ is generated by the roots - Lemma: If $F \subseteq L \subseteq K$ are fields and $K$ is a splitting field for some polynomial over $F$, it is also a splitting field for the same polynomial over $L$ - Lemma: Every polynomial $f(x)$ in $F[x]$ has a splitting field. - Lemma: A splitting field is a finite extension of $F$, every finite extension is contained in a splitting field - Theorem: Splitting theorem: Let $K$ be an extension of a field $F$ that is a splitting field of a polynomial $f(x)$ with coefficients in $F$. If an irreducible polynomial $g(x)$ with coefficients in $F$ has one root in $K$, then it splits completely in $K$. - Isomorphisms of field extensions: - Definition: $K$ and $K'$ extension fields of $F$. An $F$-isomorphism $\sigma : K \to K'$ is an isomorphism whose restriction to the subfield $F$ is the identity map. - Definition: The $F$-automorphisms of a finite extension $K$ form a group called the Galois group of $K$ over $F$, denotes $G(K/F)$. - (this is slightly different terminology to [Course - Galois Theory HT25](https://ollybritton.com/notes/uni/part-b/ht25/galois-theory/), where these are referred to as the $F$-linear automorphisms of $K$, rather than the $F$-automorphisms of $K$). - Definition: A finite extension $K/F$ is a Galois extension if $|G(K/F)| = [K : F]$ (again, slightly different but equivalent definition is given in [Course - Galois Theory HT25](https://ollybritton.com/notes/uni/part-b/ht25/galois-theory/), namely that there exists a non-constant polynomial $f \in F[x]$ such that $K$ is the splitting field of $f$). - Example: $G(\mathbb C/\mathbb R) \cong \mathbb Z_2$, and any other quadratic extension is analogous, e.g. $G(\mathbb Q(\sqrt 2) / \mathbb Q)$. - Lemma: If $K$ and $K'$ are extensions of a field $F$ and $\sigma$ is an $F$-isomorphism between them. Then if $\alpha$ is a root of $f$ in $K$, then $\sigma(\alpha)$ is a root of $f$ in $K'$. - Lemma: Suppose $K$ is generated over $F$ by elements $\alpha_1, \ldots, \alpha_n$. If $\sigma$ fixes all the generators, it's the identity. - Lemma: Let $f$ be irreducible with coefficients in $F$ and $\alpha$, $\alpha'$ are roots of $f$ in $K$ and $K'$ respectively. Then there's a unique $F$-isomorphism $\sigma : F(\alpha) \to F(\alpha')$ that sends $\alpha$ to $\alpha'$. If $F(\alpha) = F(\alpha')$, then $\sigma$ is an $F$-automorphism. - Proposition: Suppose $f$ is a polynomial with coefficients in $F$, any extension field $L/F$ contains at most one splitting field of $f$ over $F$ (and possibly zero) - Proposition: Any two splitting fields for $f$ over $F$ are isomorphic. - Fixed fields - Definition: Let $H$ be a group of automorphisms of a field $K$. The fixed field of $H$, denoted $K^H$ is $\\{\alpha \in K \mid \sigma(\alpha) = \alpha \text{ for all } \sigma \in H\\}$. - Proposition: $K^H$ is a subfield of $K$. - Proposition: $H$ is a subgroup of the Galois group $G(K/K^H)$. - Theorem: Let $H$ be a finite group of automorphisms of a field $K$ and let $F$ denote the fixed field $K^H$. Let $\beta_1$ be an element of $K$, and let $\\{\beta_1, \ldots, \beta_r\\}$ be the $H$-orbit of $B_1$. Then: - The irreducible polynomial for $\beta_1$ over $F$ is $g(x) = (x - \beta_1) \ldots (x - \beta_r)$. - $\beta_1$ is algebraic over $F$ and its degree over $F$ is equal to the order of its orbit, so the degree of $\beta_1$ over $F$ divides the order of $H$. - Definition: An extension field $K/F$ is called algebraic if every element of $K$ is algebraic over $F$. - Lemma: Let $K / F$ be algebraic that is not a finite extension of $F$. There exist elements in $K$ whose degrees over $F$ are arbitrarily large. - Theorem: Fixed Field Theorem: Let $H$ be a finite group of automorphisms of a field $K$, and let $F = K^H$ be its fixed field. Then $K$ is a finite extension of $F$, and its degree $[K : F]$ is equal to the order $|H|$ of the group. - Example: Let $K = \mathbb C(t)$ and let $\sigma, \tau$ be the automorphisms of $K$ that are the identity on $\mathbb C$ and $t \mapsto it$ and $\tau \mapsto t^{-1}$. Then $\sigma$ and $\tau$ generate the group of automorphisms $H$ that are isomorphic to the dihedral group. - Lemma: $u = t^4 + t^{-4}$ is transcendental over $\mathbb C$ - Theorem: Lüroth's theorem: Let $F$ be a subfield of $\mathbb C(t)$ of rational functions that contains $\mathbb C$ and is not $\mathbb C$ itself. Then $F$ is isomorphic to a field $\mathbb C(u)$ of rational functions. - Galois extensions - Definition: Intermediate field - Definition: When is an intermediate field proper - --- Olly Britton — https://ollybritton.com. Machine-readable index: https://ollybritton.com/llms.txt