Rubber Soul is the sixth studio album by the English rock band the Beatles. It was released on 3 December 1965 in the United Kingdom, on EMI's Parlophone label, accompanied by the non-album double A-side single "Day Tripper" / "We Can Work It Out"; the original North American version of the album, issued by Capitol Records, contained ten of the fourteen songs and two tracks withheld from the band's Help! album. Rubber Soul met with a favourable critical response and topped sales charts in Britain and the United States for several weeks; the recording sessions took place in London over a four-week period beginning in October 1965. For the first time in their career, the band were able to record an album free of concert, radio or film commitments. Referred to as a folk rock album in its Capitol configuration, Rubber Soul incorporates a mix of pop and folk musical styles; the title derives from the colloquialism "plastic soul" and was the Beatles' way of acknowledging their lack of authenticity compared to the African-American soul artists they admired.

After A Hard Day's Night in 1964, it was the second Beatles LP to contain only original material. The songs demonstrate the Beatles' increasing maturity as lyricists and, in their incorporation of brighter guitar tones and new instrumentation such as sitar and fuzz bass, the group striving for more expressive sounds and arrangements for their music; the project marked a progression in the band's treatment of the album format as an artistic platform, an approach they continued to develop with Revolver and Sgt. Pepper's Lonely Hearts Club Band; the four songs omitted by Capitol, including the February 1966 single "Nowhere Man" appeared on the North American release Yesterday and Today. Rubber Soul was influential on the Beatles' peers, leading to a widespread focus away from singles and onto creating albums of high-quality songs, it has been recognised by music critics as an album that opened up the possibilities of pop music in terms of lyrical and musical scope, as a key work in the creation of styles such as psychedelia and progressive rock.

Among its many appearances on critics' best-album lists, Rolling Stone ranked it fifth on the magazine's 2012 list "The 500 Greatest Albums of All Time". In 2000, it was voted at number 34 in the third edition of Colin Larkin's book All Time Top 1000 Albums; the album was certified 6× Platinum by the RIAA in 1997, indicating shipments of at least six million copies in the US. In 2013, Rubber Soul was certified Platinum by the BPI for UK sales since 1994. Most of the songs on Rubber Soul were composed soon after the Beatles' return to London following their August 1965 North American tour; the album reflects the influence of their month in America. Aside from setting a new attendance record when they played to over 55,000 at Shea Stadium on 15 August, the tour allowed the band to meet with Bob Dylan in New York and their longtime hero Elvis Presley in Los Angeles. Although the Beatles had released their album Help! that same month, the requirement for a new album in time for Christmas was in keeping with the schedule established in 1963 by Brian Epstein, the group's manager, George Martin, their record producer.

In their new songs, the Beatles drew inspiration from soul music the singles they heard on US radio that summer, by acts signed to the Motown and Stax record labels, from the contemporary folk rock of Dylan and the Byrds. Author Robert Rodriguez highlights the Byrds as having achieved "special notice as an American act that had taken something from the Brits, added to it sent it back". In doing so, Rodriguez continues, the Byrds had joined the Beatles and Dylan in "a common pool of influence exchange, where each act gave and took from the other in equal measure". According to music critic Tim Riley, Rubber Soul served as a "step toward a greater synthesis" of all the elements that throughout 1965 represented a "major rock'n' roll explosion", rather than just the emergence of folk rock. Citing Dylan and the Rolling Stones as the Beatles' artistic peers during this period, he says that on Rubber Soul these two acts "inspire rather than influence their sound". Two years after the start of Beatlemania, the band were open to exploring new themes in their music through a combination of their tiring of playing to audiences full of screaming fans, their commercial power, a shared curiosity gained through literature and experimentation with hallucinogenic drugs, their interest in the potential of the recording studio.

John Lennon was encouraged to address wider-ranging issues than before in his songwriting through Dylan's example. A further impetus was a discussion he had with BBC journalist Kenneth Allsop about whether, as in Lennon's 1965 book A Spaniard in the Works, his lyrics were conceived as "another form of nonsense rhyming". Author Mark Prendergast describes Rubber Soul as "the first Beatles record, noticeably drug-influenced". In Lennon's description, it was "the pot album". Recording for Rubber Soul began on 12 October 1965 at EMI Studios, with final production and mix down taking place on 15 November. During the sessions, the Beatles focused on fine-tuning the musical arrangement for each song, an approach that reflected the growing division between the band as a live act and their ambitions as recording artists; the album was one of the first projects that Martin undertook after leaving EMI's staff and co-founding Associated Independent Recording. Martin described Rubber Soul as "the first album to present a new, growing Beatles to the world", adding: "For the first time we began to think of albums as art on their own, as complete entities."

This was the final Beatle album that recording engineer Norman Smith worked on before

Rennie Airth is a South African novelist who resides in Italy. Airth has worked as foreign correspondent for the Reuters news service, his works include Snatch!, Once A Spy, a series of murder mysteries set in England between 1921 and 1949 featuring Detective Inspector John Madden of Scotland Yard. The first of these, River of Darkness, won the Grand Prix de Littérature Policière for best international crime novel in 2000 and was nominated for Edgar and Macavity awards in the States. Airth found inspiration for that tale in a scrapbook about his uncle, a soldier killed in World War I. A sequel, The Blood-Dimmed Tide, was published in 2003, a third book, The Dead of Winter, in 2009. Although Airth intended to write a trilogy about Madden, in 2014 he produced a fourth entry in the series, The Reckoning, followed that with The Death of Kings. Snatch Once A Spy Detective Inspector John Madden River of Darkness The Blood-Dimmed Tide The Dead of Winter The Reckoning The Death of Kings The Decent Inn of Death The story behind The Reckoning - Essay by Rennie Airth on Upcoming4.me

Hilbert's Nullstellensatz is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of a branch of mathematics, it relates algebraic sets to ideals in polynomial rings over algebraically closed fields. This relationship was discovered by David Hilbert who proved the Nullstellensatz and several other important related theorems named after him. Let k be a field and K be an algebraically closed field extension, consider the polynomial ring k and let I be an ideal in this ring; the algebraic set V defined by this ideal consists of all n-tuples x = in Kn such that f = 0 for all f in I. Hilbert's Nullstellensatz states that if p is some polynomial in k that vanishes on the algebraic set V, i.e. p = 0 for all x in V there exists a natural number r such that pr is in I. An immediate corollary is the weak Nullstellensatz: The ideal I ⊂ k contains 1 if and only if the polynomials in I do not have any common zeros in Kn, it may be formulated as follows: if I is a proper ideal in k V cannot be empty, i.e. there exists a common zero for all the polynomials in the ideal in every algebraically closed extension of k.

This is the reason for the name of the theorem, which can be proved from the'weak' form using the Rabinowitsch trick. The assumption of considering common zeros in an algebraically closed field is essential here. With the notation common in algebraic geometry, the Nullstellensatz can be formulated as I = J for every ideal J. Here, J denotes the radical of J and I is the ideal of all polynomials that vanish on the set U. In this way, we obtain an order-reversing bijective correspondence between the algebraic sets in Kn and the radical ideals of K. In fact, more one has a Galois connection between subsets of the space and subsets of the algebra, where "Zariski closure" and "radical of the ideal generated" are the closure operators; as a particular example, consider a point P = ∈ K n. I =. More I = ⋂ ∈ V. Conversely, every maximal ideal of the polynomial ring K is of the form for some a 1, …, a n ∈ K; as another example, an algebraic subset W in Kn is only if I is a prime ideal. There are many known proofs of the theorem.

One proof uses Zariski's lemma, which asserts that, if a field is finitely generated as an associative algebra over a field k it is a finite field extension of k. Here is a sketch of this proof. Let A = k, I an ideal of A and V the common zeros of I in