Norms, Traces and Discriminants
Handbook guide to norms, traces and discriminants with core definitions, structural results, reasoning methods and verification checks.
How the topic fits together
Norms and Traces
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
The Discriminant
This source section supplies the definitions, constructions and formal results used in this article. The sequence of results is preserved so later claims are not detached from their prerequisites.
Core definitions and structural results
The following cards retain the mathematical content of the source while condensing long proofs into verification strategies. Numerical examples are treated as examples, not universal requirements.
Definitions and Comments
If E/F is a field extension of finite degree n, then in particular, E is an n-dimensional vector space over F, and the machinery of basic linear algebra becomes available. If x is any element of E, one can study the F-linear transformation m(x) given by multiplication by x, that is, m(x)y = xy. Define the norm and the trace of x, relative to the extension E/F, as N[E/F](x) = det m(x) and T[E/F](x) = trace m(x). We will write N(x) and T(x) if E/F is understood.
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Example
Let E = C and F = R. A basis for C over R is {1, i} and, with x = a + bi, one has (a + bi)(1) = a(1) + b(i) and (a + bi)(i) = −b(1) + a(i). Thus A(a + bi) = a −b b a . The norm, trace and characteristic polynomial of a + bi are N(a + bi) = a2 + b2, T(a + bi) = 2a, char(a + bi) = X2 −2aX + a2 + b2.
Proof / verification strategy: Treat this as an illustration of the surrounding definitions. Recompute the stated relations directly and keep the numerical or structural choices local to the example.
Lemma
If E is an extension of F and x ∈E, then N(x), T(x) and the coefficients of char(x) belong to F. If a ∈F, then N(a) = an, T(a) = na, and char(a) = (X −a)n.
Proof / verification strategy: Work through generators and minimal polynomials, use basis/degree information, and track how field automorphisms act on roots and intermediate fields.
Proposition
char[E/F](x) = [min(x, F)]r where r = [E : F(x)].
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Corollary
Let [E : F] = n, and [F(x) : F] = d. Let x1, . . . , xd be the roots of min(x, F) in a splitting field (counting multiplicity). Then N(x) = ( d i=1 xi)n/d, T(x) = n d d i=1 xi and char(x) = [ d i=1 (X −xi)]n/d.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Proposition
Let E/F be a separable extension of degree n, and let σ1, . . . , σn be the distinct F-monomorphisms of E into an algebraic closure of E, or equally well into a normal extension L of F containing E. Then T[E/F](x) = n i=1 σi(x) and N[E/F](x) = n i=1 σi(x). Consequently, T(ax + by) = aT(x) + bT(y) and N(xy) = N(x)N(y) for x, y ∈E, a, b ∈F.
Proof / verification strategy: Work through generators and minimal polynomials, use basis/degree information, and track how field automorphisms act on roots and intermediate fields.
Corollary (Transitivity of Trace and Norm)
(Transitivity of Trace and Norm) If F ≤K ≤E, where E/F is finite and separable, then T[E/F] = T[K/F] ◦T[E/K] and N[E/F] = N[K/F] ◦N[E/K].
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Corollary
If E/F is a finite separable extension, then T[E/F](x) cannot be 0 for all x ∈E.
Proof / verification strategy: Work through generators and minimal polynomials, use basis/degree information, and track how field automorphisms act on roots and intermediate fields.
The Basic Setup For Algebraic Number Theory
The Basic Setup For Algebraic Number Theory Let A be an integral domain with quotient field K, and let L be a finite separable extension of K. Let B be the set of elements of L that are integral over A, that is, B is the integral closure of A in L. The diagram below summarizes all the information. L B | | K A In the most important special case, A = Z, K = Q, L is a number field, and B is the ring of algebraic integers of L.
Proof / verification strategy: Work through generators and minimal polynomials, use basis/degree information, and track how field automorphisms act on roots and intermediate fields.
Proposition
If x ∈B, then the coefficients of char[L/K](x) and min(x, K) are integral over A. In particular, T[L/K](x) and N[L/K](x) are integral over A, by (7.3.2). If A is integrally closed, then by (7.3.3), the coefficients belong to A.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Definition
The discriminant of the n-tuple x = (x1, . . . , xn) of elements of L is D(x) = det(T[L/K](xixj)). Thus we form a matrix whose ij element is the trace of xixj, and take the determinant of the matrix. By (7.3.3) and (7.3.10), D(x) belongs to K and is integral over A, hence belongs to A if A is integrally closed. The discriminant behaves quite reasonably under linear transformation:
Proof / verification strategy: Use this as a definition checklist: identify the ambient object, test each stated condition separately, then compare examples and non-examples without assuming later theorems.
Lemma
If y = Cx, where C is an n by n matrix and x and y are n-tuples written as column vectors, then D(y) = (det C)2D(x).
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Lemma
Let σ1, . . . , σn be the K-embeddings of L into an algebraic closure of L, as in (7.3.6). Then D(x) = [det(σi(xj))]2. Thus we form the matrix whose ij element is σi(xj), take the determinant and square the result.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Proposition
If x = (x1, . . . , xn), then the xi form a basis for L over K if and only if D(x) ̸= 0.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Proposition Assume that L = K(x), and let f be the minimal polynomial of x over
Assume that L = K(x), and let f be the minimal polynomial of x over K. Let D be the discriminant of the basis 1, x, x2, . . . , xn−1 for L over K. Then D is the discriminant of the polynomial f.
Proof / verification strategy: Work through generators and minimal polynomials, use basis/degree information, and track how field automorphisms act on roots and intermediate fields.
Corollary Under the hypothesis of (7.4.5),
Under the hypothesis of (7.4.5), D = (−1)( n 2)N[L/K](f ′(x)) where f ′ is the derivative of f.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Lemma There is a basis for L/K consisting entirely of elements of B.
There is a basis for L/K consisting entirely of elements of B.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Theorem
Suppose one has a nondegenerate symmetric bilinear form on an ndimensional vector space V , written for convenience using inner product notation (x, y). If x1, . . . , xn is any basis for V , then there is a basis y1, . . . , yn for V , called the dual basis referred to V , such that (xi, yj) = δij = 1, i = j; 0, i ̸= j. This is a standard (and quite instructive) result in linear algebra, and it will be developed in the exercises.
Proof / verification strategy: Start from the stated definitions, prove each required implication or inclusion separately, and use the immediately preceding structural result where it shortens the argument.
Theorem
If A is a principal ideal domain, then B is a free A-module of rank n.
Proof / verification strategy: Translate the element statement into ideals or quotient rings, establish both inclusions or implications, and use the strongest ring hypothesis actually available.
Quick-reference relationships
Problem-solving workflow
Identify the ambient domain and extension
State the base ring or field and the integral elements under discussion.
Translate arithmetic into ideals or linear maps
Norms, traces, discriminants and ideal factorisation encode arithmetic structurally.
Track divisibility and integrality
Do not confuse being algebraic with being integral, or element factorisation with ideal factorisation.
Use local information where appropriate
Valuations and p-adic ideas measure divisibility using a topology different from the usual absolute value.
Check finiteness hypotheses
Finite generation, Noetherian conditions and finite extensions are frequently essential.
Verify with quadratic or integer examples
Use small extensions and ideals to test signs, degrees and multiplicities.
Worked-solution emphasis from the supplied source
The supplied worked solutions for this section repeatedly test polynomial, root, degree, basis, field, prime, factor, trace. These checks are used here as verification themes rather than copied as answer text.
Common mistakes and boundary conditions
- Using the ordinary absolute value when a p-adic valuation is intended.
- Confusing element factorisation with ideal factorisation.
- Assuming integrality without a monic polynomial relation.
Verification checklist
- State the ambient algebraic structure and operation before applying a theorem.
- Record every hypothesis that controls the result: finiteness, commutativity, normality, primality, separability, exactness or other section-specific conditions.
- Distinguish a definition from a theorem that follows from it.
- Check whether a map is well-defined before using its kernel, image, inverse or induced map.
- Use a concrete example only as a check; do not promote an illustrative value or pattern to a universal rule.
- When a quotient, localisation or extension is constructed, identify the canonical map and what becomes equal, invertible or fixed.
Source coverage map
| Source section | Subject | PDF pages analysed |
|---|---|---|
| 7.3 | Norms and Traces | 135–138 |
| 7.4 | The Discriminant | 139–141 |
Related Mathematics pages
Source note: synthesised from the supplied abstract-algebra PDF. The complete 298-page file, including diagrams and worked solutions, was reviewed. Source-identifying author and bibliographic personal details are intentionally omitted. Formal proofs are condensed; the page does not claim requirements or values not supported by the supplied mathematics.