Cyclotomic field

A cyclotomic field is generated by a root of unity. If $K\subseteq \mathbb{C}$ and ${\zeta }_n$ is primitive, every $K$-conjugate of ${\zeta }_n$ is a power ${\zeta }_n^a$ and already lies in $K({\zeta }_n)$. Thus $K({\zeta }_n)/K$ is Galois, and its Galois group embeds in $(\mathbb{Z}/n\mathbb{Z}{)}^{\times }$, so it is abelian.

A Gaussian period is a sum of roots of unity indexed by a multiplicative coset in the group of nonzero residues modulo a prime number. Such sums describe subfields of a cyclotomic field and occur as values of induced characters of certain semidirect products.

With $\zeta =e^{2\pi i/7}$, put $\alpha =\zeta +{\zeta }^2+{\zeta }^4$ and $\beta ={\zeta }^3+{\zeta }^5+{\zeta }^6$. These Gaussian periods obey $\beta =\overline{\alpha }$, $\alpha +\beta =-1$, and ${\alpha }^2=\alpha +2\beta$. Consequently ${\alpha }^2+\alpha +2=0$. The positive imaginary part of $\alpha$ fixes $\alpha =(-1+i\sqrt{7})/2$ and $\beta =(-1-i\sqrt{7})/2$. They are the nonidentity kernel values of the two degree-three irreducible characters of the nonabelian group of order 21.

The Cyclotomic character of a finite cyclotomic extension and the Chinese remainder theorem give the displayed Galois group. It has three order-two subgroups and three order-four subgroups; the latter are kernels of its three nonzero group homomorphisms to $C_2$. By the Fundamental theorem of Galois theory, its proper intermediate fields strictly above $\mathbb{Q}$ are the three quadratic fieldsand the three quartic fieldsThe first quartic has cyclic Galois group of order four and unique quadratic subfield $\mathbb{Q}(\sqrt{5})$; the second is a biquadratic extension, so they differ. The last is real whereas the others are not, by the maximal real subfield of a cyclotomic field formula. Also $\sqrt{-3}+\sqrt{5}$ generates the second quartic: if $s=\sqrt{-3}+\sqrt{5}$, then $\sqrt{-3}=(s-8/s)/2$ and $\sqrt{5}=(s+8/s)/2$.

For $m\ge 3$, put $\eta ={\zeta }_m+{\zeta }_m^{-1}$. Since $\eta$ is real and ${\zeta }_m$ is nonreal but satisfies $T^2-\eta T+1=0$, the cyclotomic field $\mathbb{Q}({\zeta }_m)$ has degree two over $\mathbb{Q}(\eta )$. Complex conjugation fixes $\eta$ and is nontrivial. The Fundamental theorem of Galois theory therefore identifies its fixed field with $\mathbb{Q}(\eta )$. That fixed field is exactly $\mathbb{Q}({\zeta }_m)\cap \mathbb{R}$, proving maximality among real subfields.

For a primitive fifth root ${\zeta }_5$, the element $s={\zeta }_5+{\zeta }_5^{-1}$ satisfies $s^2+s-1=0$. Thus the fixed field of complex conjugation is

For a primitive $m$th root of unity ${\zeta }_m$, every Galois automorphism sends ${\zeta }_m$ to a unique ${\zeta }_m^a$ with $a$ invertible modulo $m$. The assignment $\sigma \mapsto a$ is a group homomorphism, since composition multiplies the exponents, and is injective because ${\zeta }_m$ generates the cyclotomic field. The irreducibility of cyclotomic polynomials gives degree $\phi (m)$, equal to the order of the target, so the map is an isomorphism. The cyclotomic character on an absolute Galois group factors through this finite quotient and generally has a nontrivial kernel.

A cyclotomic integer is an element of $\mathbb{Z}[{\zeta }_n]$, where ${\zeta }_n$ is a root of unity. For a primitive $n$-th root, this is the full ring of integers of the cyclotomic field $\mathbb{Q}({\zeta }_n)$.

For the cyclotomic field $\mathbb{Q}({\zeta }_n)$, arithmetic Frobenius automorphisms at primes $\ell \nmid n$ send ${\zeta }_n$ to ${\zeta }_n^{\ell }$: reduction distinguishes the nth roots of unity, and Frobenius raises residues to their $\ell$th powers. Multiplicativity therefore sends a fractional ideal with positive rational generator $a$, prime to $n$, to the exponent $a\mathrm{mod}n$. Its kernel consists exactly of positive generators congruent to one modulo $n$. This proves Artin reciprocity for the ray class group of modulus $n\infty$ without assuming the general reciprocity theorem.

Cyclotomic units include roots of unity and suitable ratios $(1-{\zeta }^a)/(1-\zeta )$ in cyclotomic fields. Compatible roots $\zeta$ make these ratios norm-compatible along cyclotomic towers. Their closed norm-limit module is a concrete submodule of global units, and its local images generate the power series in the Iwasawa main conjecture.

For $1\le k<p$, choose $l$ with $kl\equiv 1(\mathrm{mod}p)$. The geometric sum $1+{\zeta }_p^k+\cdots +{\zeta }_p^{k(l-1)}$ is the inverse of $u_k$. Hence these are units, and their residues cover ${\mathbb{F}}_p^{\times }$. In the cyclotomic Milnor square, this makes the $K_1$ map onto the common field surjective and eliminates its connecting obstruction to $K_0$.

For a real cyclotomic field of odd prime-power conductor, the index of its cyclotomic-unit group has the same p-part as its ideal class number. The analytic class number formula compares the regulator of all units with the explicit cyclotomic-unit regulator. The equality here is stated at the p-part level, so harmless prime-to-p normalization factors do not affect it.

Half exponents mean the inverse of two in the odd-order root-of-unity group. For $p\nmid ab$, the ratio is a real global unit and is compatible under the norms in the p-power tower. Its Coleman power series is $(1+T{)}^{(b-a)/2}(1-(1+T{)}^a)/(1-(1+T{)}^b)$, with constant term $a/b$. Its higher logarithmic derivatives vanish at odd indices and equal $(a^k-b^k)B_k/k$ at even indices, using $B_1=-1/2$.

A cyclotomic Euler system consists of units in auxiliary cyclotomic extensions satisfying compatible norm relations. Reducing at carefully chosen auxiliary primes and descending these relations bounds class-group modules. Combined with global unit-index formulas, this supplies an alternative route to the Iwasawa main conjecture.

Fix a finite excluded set $S$ containing two, and let $W_S$ contain the roots of unity of orders coprime to every prime in $S$. A root-of-unity Euler system is a nonzero algebraic-valued function $\Phi$ on $W_S$ that is even and Galois equivariant, satisfies ${\prod }_{{\rho }^q=1}\Phi (\rho \zeta )=\Phi ({\zeta }^q)$ for $q\notin S$, and satisfies $\Phi (\rho \zeta )\equiv \Phi (\zeta )(\mathrm{mod}\mathfrak{q})$ when the order of $\zeta$ is prime to $q$, for every prime $\mathfrak{q}$ over $q$. Values are required to be locally integral units where the congruence is used. These distribution and congruence properties provide auxiliary cyclotomic extensions, beyond a single norm-compatible p-power sequence.

Let $q\nmid m$ be outside the excluded primes, let $\zeta$ have order $m$, and let $\rho$ be a primitive qth root of unity. Galois equivariance identifies the conjugates with $\Phi ({\rho }^i\zeta )$ for $1\le i\le q-1$. The distribution axiom includes the missing $i=0$ factor, so the field norm is $\Phi ({\zeta }^q)/\Phi (\zeta )$. Arithmetic Frobenius sends $\zeta$ to ${\zeta }^q$, giving the displayed factor. At higher powers of an already ramified prime all q translates occur instead. These are different norm relations; norm compatibility in just a p-power tower does not supply the auxiliary-prime relation by itself.

For a cyclic group of order $m$ with generator $\sigma$, let $N={\sum }_{i=0}^{m-1}{\sigma }^i$. Subtracting adjacent coefficients and using ${\sigma }^m=1$ proves the displayed group-ring identity. If $p^r\mid m$, then $(\sigma -1)D\equiv -N(\mathrm{mod}{p}^r)$. Together with the norm relations of a root-of-unity Euler system, this makes derivative classes invariant modulo p-power powers and permits Kummer descent, with roots-of-unity corrections where necessary. The congruence axiom controls the associated residue and valuation maps.

The divisibility is between generators of characteristic ideals. Auxiliary-prime derivative operators applied to a root-of-unity Euler system produce Kummer classes with controlled valuations. The congruence axiom identifies the valuation at a new auxiliary prime with a residue symbol of the preceding class. The Chebotarev density theorem chooses prime representatives of desired ideal classes, and iterating bounds their elementary divisors by the index of the cyclotomic unit. Passing to the norm tower gives the displayed divisibility, with finite errors removed at height-one primes. The reverse divisibility requires the global index and norm-defect comparison.

Every field $\mathbb{Q}({\zeta }_{59^r})$ contains $\mathbb{Q}(\sqrt{-59})$ and is totally ramified over it at the prime above $59$. The class number divisibility under total ramification therefore shows that its class number is divisible by three. Its ring of integers is not a principal ideal domain. The degrees $58\cdot 59^{r-1}$ strictly increase, giving infinitely many distinct examples.

The prime $\ell$ is totally ramified in $\mathbb{Q}({\zeta }_{{\ell }^r})$, whose degree is $\phi ({\ell }^r)$. The displayed equality is an equality of ideals in its ring of integers of a number field, and the residue field is ${\mathbb{F}}_{\ell }$.

For an odd prime $\ell$, the unique quadratic subfield of $\mathbb{Q}({\zeta }_{\ell })$ is the displayed field. A quadratic Gauss sum squares to $(-1{)}^{(\ell -1)/2}\ell$, and the cyclic Galois group has a unique subgroup of index two.

For a prime $p$, the cyclotomic polynomial of a primitive $p$th root of unity isIt is irreducible over $\mathbb{Q}$, so adjoining one root gives a field of degree $p-1$.

For an odd prime $p$ and a primitive $p$th root ${\zeta }_p$,

The roots of unity in $\mathbb{Q}({\zeta }_n)$ form a cyclic group of order

For a primitive seventh root of unity ${\zeta }_7$,where ${\sigma }_a({\zeta }_7)={\zeta }_7^a$. Its four intermediate fields correspond to the four subgroups of the cyclic group $C_6$.

The unique quadratic subfield of $\mathbb{Q}({\zeta }_7)$ isThe conjugate period is $-1-\eta$, and

The unique cubic subfield is the maximal real subfieldandIt is the fixed field of complex conjugation.

The Galois group of $\mathbb{Q}({\zeta }_{11})/\mathbb{Q}$ isIts only subfield degrees over $\mathbb{Q}$ are $1,2,5,10$, and every subfield is Galois because this group is abelian.

The unique quadratic subfield of $\mathbb{Q}({\zeta }_{11})$ is

The fixed field of complex conjugation in $\mathbb{Q}({\zeta }_{11})$ is the degree-five field $\mathbb{Q}({\zeta }_{11}+{\zeta }_{11}^{-1})$.

For a field $K$ inside a fixed algebraic closure, the maximal cyclotomic extension is the union of the fields $K({\zeta }_n)$ generated by roots of unity.

The maximal cyclotomic extension of $\mathbb{R}$ is $\mathbb{C}$, since adjoining a primitive fourth root of unity adjoins $i$.

The maximal cyclotomic extension of $\mathbb{Q}$ is a proper subfield of $\overline{\mathbb{Q}}$. Every finite subextension inside a cyclotomic field is Galois and abelian, whereas $\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}$ is not normal.

The maximal cyclotomic extension of ${\mathbb{F}}_p$ is ${\overline{\mathbb{F}}}_p$. A primitive $(p^m-1)$th root of unity generates ${\mathbb{F}}_{p^m}$ over ${\mathbb{F}}_p$, and these finite fields exhaust the algebraic closure.