We briefly describe our results. In Section 3, we focus our attention on the case of quadratic fields. Let z be a fixed primitive pth root of unity in Cp. The symbol.

Let C be the group of real cyclotomic units of Q z ; i. Let D be the g.

We are in position to prove the following theorem. Theorem 2. Let p be the unique prime of OF above p. Note that a 2 ] 1. It remains to apply [1, Corollary 5. L Remarks. Our result in this case has already been obtained by Kolyvagin see [7].

We will need the following result in the following. If P z is orthogonal to C for the norm residue symbol, then ab — a 2 b —1 mod p , mod p. Set UK.

### Norm Residue Symbol and the First Case of Fermat's Equation

Then, by [1, Corollary 4. We have dim B [ r. Note that there exists an integer i, 1 [ i [ r, such that bi — 0 mod p. L Theorem 3. Let F be a quadratic field. Note that we can assume x2 — y2 mod p. By the basic argument of the second section, P z and G z are orthogonal to C for the norm residue symbol. This is a contradiction since p — 2 mod 3. This implies a 2 — 1 mod p , which is a contradiction.

L Now, we consider the case of real quadratic fields. Let F be a real quadratic field and let q be the nontrivial Dirichlet character associated to F. We assume that p does not divide d F. Note that Ernvall see [2] has proved that there exist infinitely many q-irregular primes. Theorem 3. This will return an identical result when given K as input again. Test that trac ticket is fixed:. This is the product of all finite primes where the Hilbert symbol is If a and b are non-zero and P is unspecified, returns 1 if the equation has a solution in self and -1 otherwise.

That the latter two are unsolvable should be visible in local obstructions. For the first, this is a prime ideal above For the second, the ramified prime above Check that the bug reported at trac ticket has been fixed:. The implementation is following algorithm 3. We note that class and unit groups are computed using the generalized Riemann hypothesis. If it is false, this may result in an infinite loop.

Nevertheless, if the algorithm terminates the output is correct. If B is not None, return the images of the vectors in B as the columns instead. Then the Minkowski embedding is given by. Return a field isomorphic to self with a better defining polynomial if possible, along with field isomorphisms from the new field to self and from self to the new field. Return optimized representations of many but not necessarily all!

By default, this returns the set of real places as homomorphisms into RIF first, followed by a choice of one of each pair of complex conjugate homomorphisms into CIF. Return all subfields of self of the given degree, or of all possible degrees if degree is 0. The subfields are returned as absolute fields together with an embedding into self. For the case of the field itself, the reverse isomorphism is also provided.

The command CyclotomicField n creates the n-th cyclotomic field, obtained by adjoining an n-th root of unity to the rational field. Note in the example above that the way zeta is computed using sin and cosine in MPFR means that only the prec bits of the number after the decimal point are valid.

### Categories

Return all embeddings of this cyclotomic field into the approximate complex field with precision prec. If you want bit double precision, which is faster but less reliable, then do self. Return data defining a functorial construction of self. Returns the discriminant of the ring of integers of the cyclotomic field self, or if v is specified, the determinant of the trace pairing on the elements of the list v.

## Artin reciprocity law - Wikipedia

Uses the formula for the discriminant of a prime power cyclotomic field and Hilbert Theorem 88 on the discriminant of composita. Check trac ticket :. Return all embeddings of this cyclotomic field into the approximate real field with precision prec. Return r1, r2 , where r1 and r2 are the number of real embeddings and pairs of complex embeddings of this cyclotomic field, respectively.

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If there is no such element, raise a ValueError. WithEqualityById , sage. An example in a relative extension see trac ticket :. Our generators should have the correct parent trac ticket :. Return self as an absolute number field. Mostly for internal use. Only implemented at archimedean places, and then only if an embedding has been fixed. This is only well-defined for fields contained in CM fields i.

Recall that a CM field is a totally imaginary quadratic extension of a totally real field. For other fields, a ValueError is raised. Return all homomorphisms of this number field into the approximate complex field with precision prec. This always embeds into an MPFR based complex field.

## Class Field Theory. Field Extensions

If you want embeddings into the bit double precision, which is faster, use self. Return the possible composite number fields formed from self and other. In all other cases, all possible composite number fields are returned. A particular compositum is selected, together with compatible maps into the compositum, if the fields are endowed with a real or complex embedding:.

This is just one of four embeddings of Q1 into F :. This is exactly the same as self.

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- Norm Residue Symbol and the First Case of Fermat's Equation!

Returns the discriminant of the ring of integers of the number field, or if v is specified, the determinant of the trace pairing on the elements of the list v. Return a list of elements of norm n. Extension fields with given defining data are unique trac ticket :. Here we show how to factor Gaussian integers up to units.